LECTURES ON DIOPHANTINE APPROXIMATIONS Part 1: g-adic numbers and Roth's theorem
By
KURT MAHLER
Prepared from the notes by R.P. Bambah of my lectures given at the University of Notre Dame in the Fall of 1957
Copyright 1961 UNIVERSITY OF NOTRE DAME
FHOTOLITHOFRINTED BY GUSHING - MALLOY, ING. ANN ARBOR, MICHIGAN, UNITED STATES OF AMERICA 1961
Dedicated in Gratitude to
L. J. Mordell
My interest in two of the main subjects of these lectures goes back to my student days at Frankfurt (1923-25) and Gottingen (1925-33). Prom C. L. Siegel I learned of Thue's theorem and its improvements and generalisations; and Emmy Noether introduced me to the theory of p-adic numbers. I combined these two ideas in 1931 when I found an analogue of the Thue- Siegel theorem that involved both real and p-adic algebraic numbers. In later work I repeatedly came back to such problems, and already in 1934-36 I gave a course on Diophantine approximations at the University of Groningen dealing with problems that simultaneously involved the real and p-adic fields. After the war, E. Lutz published her very beautiful little book on Diophantine approximations in the p-adic field. But she considered alone the case of numbers in one fixed p-adic field. In 1955, K. F. Roth obtained his theorem on the rational approximations of a real algebraic number. It was immediately clear that his method should also work for p-adic algebraic numbers, for Roth's method could clearly be combined with that of my old papers. Some interesting work of this kind was in fact carried out by D. Ridout, a student of Roth. In the second part of these lectures I shall try to go rather further in this direction. As the proofs will show, the p-adic numbers, and more generally the g-adic numbers form an important tool in these investigations; but one form of the final result will be again free of such numbers and will state a property of rational numbers only. The first part of these lectures has mainly the purpose of acquainting the reader with the theory of p-adic and g-adic numbers. It gives a short account of the theory of valuations, and I have found it convenient to discuss also the slightly more general theory of pseudovaluations because it leads in a very natural way to Hensel's g-adic numbers. The results in Chapters 3 and 4 serve chiefly as examples, but have perhaps also a little interest in themselves. The whole second part, as well as two short appendices, deal with a very general g-adic form of Roth's theorem. As the proof is rather involved, all details are given, and I have also tried to explain the reasons behind the different steps of the proof. The most original part of Roth's proof consists in a very deep theorem, here called Roth's Lemma. It states that, under certain conditions, a polynomial in a large number of variables cannot have a multiple zero of too high an order. Since Roth's Lemma is essentially a theorem on the singularities of an algebraic manifold, perhaps the methods of algebraic geometry may finally enable one to obtain a simpler proof and a stronger result (say, with the upper bound 2*n+112"^"1^ in Roth's Lemma replaced by something like tm"c). It would then become possible to improve the theorem by Cugiani given in the appendix. Another possible approach to Roth's Lemma is from the theory of
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LECTURES ON DIOPHANTINE APPROXIMATIONS
interpolation, or from Minkowski's theorem on the successive minima of convex bodies. As far as I know, neither of these methods has yet been applied to the problem. The method of Thue-Siegel-Roth has one fundamental disadvantage, that of its non- effectiveness. The proof is entirely non- constructive, and by its very nature does not lead to any upper bounds for possible solutions. Only in some very special cases effective methods are known and there are due to Skolem and Gelfond. Certainly the methods and results of the second part of these lectures are not the last word on the subject, and entirely new ideas are called for. So far one has not succeeded even in proving that there exist real algebraic numbers a for which the inequality
has infinitely many rational solutions Q , however small 6 > o is chosen; or that there are real algebraic numbers a. at least of the third degree for which this is not the case. I should be well pleased if these lectures succeed in convincing the reader that the whole subject is as yet in a very unsatisfactory state. Essentially in the form as printed here, I gave these lectures during the , Fall term of 1957 at the University of Notre Dame, and I wish to express my thanks to the authorities of Notre Dame for the invitation to work and lecture at this very pleasant place. My particular thanks are due to all who attended my lectures and especially to Professors Ross, Lewis, Skolem, and Bambah. In many talks with these colleagues, on the way to lunch after the lectures, much became clearer, and simpler proofs were found. I am especially indebted to R. P. Bambah who undertook the thankless task of taking down the lectures, and to T. Murphy who checked this manuscript for errors. After the work of more than a year Bauebah's notes have now at last helped me to complete this book. As the title already suggests, I hope to continue these lectures at a later date. A second part will probably deal with applications of the geometry of numbers to Diophantine approximations in p-adic fields. Concluded on March 12, 1959 Mathematics Department, Manchester University.
LIST OF NOTATIONS The meanings of letters and symbols will usually be clear from the context. In general, small Latin letters denote rational numbers, small Greek letters denote real or p-adic numbers, and capital Greek letters denote gadicor g*-adic numbers. By F, P, Pp, Pg, and Pg* we mean the fields of rational, real, and p-adic numbers, and the rings of g-adic, and g*-adic numbers, respectively. The symbols '<*!> l^olp, Wig, and U*|g* stand for the absolute value of the real number a, the p-adic value of the padic number ao, the g-adic value of the g-adic number A, and the g*-adic value of the g*-adic number A*, respectively. Here lao'p is normed by the formula
The integer g ^ 2 always has the prime factorisation ei er g=Pl ---Pr > where pj , . . . , pr are distinct primes, and ej , . . . , er are positive integers. If, for j = 1, 2, . . . , r, the g-adic number A has the pj-adic component o?j, we write and then logg log PI
logg er log pr
max
Thus, in particular,
A g*-adic number A* has, in addition to the pj-adic components «j , also a real component a . We write <*i , . . . , otr) = (a,A) where A^ai , . . . , ar). Then U*lg* =max(|al, Ulg). For rational integers a, b, m =j= o the congruence a s b (mod m) means, as usual, that a - b is divisible by m. Instead of a = o (mod m) we write m la. The symbol (a, b, . . . , f) means the greatest common divisor of the rational integers a, b, . . . , f , except on certain occasions when the same symbol is used to denote an ordered set of numbers. If o? is a real number, [a] always denotes the integral part of a, i. e. the integer a for which a 4 a < a + 1. vii
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LECTURES ON DIOPHANTINE APPROXIMATIONS
A formula like a e S means that a is an element of the set S. If Sj and 82 are sets, Si u 82 is their union and Si n 82 their intersection (i.e. SiU 82 consists of all a that are elements of at least one of the sets, and Si n 82 of all a that -are elements of both sets.) The signs u 8^ and n \ are used for k k the union and intersection of any number of sets.
LIST OF CONTENTS Preface
v
List of Notations Fart 1: p-adic and g-adic Numbers, and Their Approximations I. Valuations and pseudo-valuations 1. 2. 3. 4.
II.
III.
Valuations and pseudo-valuations The p-adic valuations of P A further example Valuations and pseudo-valuations derived from given ones 5. Bounded sequences, fundamental sequences, and null sequences 6. The ring (K}w and the ideal Jt 7. The residue class ring K^ 8. The completion of a field with respect to a valuation 9. The limit notation 10. The continuation of w(a) onto K^ 11. The elements of K lie dense in Kw 12. Fundamental sequences in Kw 13. Equivalence of valuations and pseudo-valuations 14. The valuations and pseudo-valuations of P 15. Independent pseudo-valuations 16. The decomposition theorem 17. Convergent infinite series The p-adic, g-adic, and g*-adic series 1. Notation 2. The ring Ig and the ideal 9 3. The residue class ring Ig/g 4. Systems of representatives 5. Series for g-adic numbers 6. Series for g*-adic numbers 7. Sequences that converge with respect to all valuations of r A test for algebraic or transcendental numbers 1. 2. 3. 4. 5. 6.
Notation , The minimum polynomial of an algebraic number An algebraic identity Inequalities for algebraic numbers A theorem on linear forms On a system of both real and p-adic linear forms ix
vii 1 3 4 5 6 7 9 11 12 13 13 14 15 16 17 18 20 21 24 26 27 29 30 31 32 36 40 41 42 42 43 45 48 50
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LECTURES ON DIOPHANTINE APPROXIMATIONS 7. Polynomials F(x) for which o>(F (a)) is small 8. A necessary and sufficient condition for transcendency
IV.
Continued fractions 1. The continued fraction algorithm in the real case 2. The convergents of the continued fraction for <*0 3. The distinction between rational and irrational numbers 4. Inequalities for J Qj^ a0— Pfc| 5. The convergents as best approximations 6. The rational approximations of g-adic integers 7. The continued fraction algorithm for a g-adic integer 8. Two numerical examples . .' 9. Final remarks to the g-adic algorithm 10. The continued fraction algorithm for g*-adic numbers Part 2: Rational Approximations of Algebraic Numbers. . ' The Problem and Its History V. 1. Introduction 2. Linear dependence and independence 3. Generalized Wronski determinants 4. The case of functions of one variable 5. The general case 6. An identity 7. Majorants for U, V, and W 8. The index of a polynomial 9. The upper bound 0m (a; H! , . . . , Hm; ra , . . . , rm) 10. An upper bound for 0j (a; r; H) 11. The property TM 12. A recursive inequality for 8m. 1 13. A recursive inequality for 0m. II. 14. Proof of Roth's Lemma VI. The Approximation Polynomial
VII.
53 55 58 58 59 60 62 63 63 64 67 69 69 73 77 77 78 79 80 82 85 87 89 90 90 \ 92 93 96 98
1. The aim 2. The powers of an algebraic number 3. A lemma by Schneider 4. The construction of A(xi , . . . , xm). 1 5. The construction of A(XI , . . ., xm). II 6. The construction of A(XJ , . . . , xm). m The First Approximation Theorem
98 98 99 101 102 104 107
1. 2. 3. 4. 5.
107 109 112 113 116
The properties Ad, B, and C The selection of the parameters Application of Theorems 1 and 2 Upper bounds for | A(y| An upper bound for | A(i)|
CONTENTS 6. 7. 8. 9. 10. 11. 12. 13. VIII.
IX.
An upper bound for | D(j)| Lower bounds for | N/y | Conclusion of the prooi The first form of the First Approximation Theorem Polynomials in a field with a valuation Two applications of Lemma 1 The property A^ The second form of the First Approximation Theorem The Second Approximation Theorem 1. The two forms of the theorem 2. The Theorem (2,11) implies the Theorem (2,1) 3. The Theorem (2,1) implies the Theorem (2,11) 4. The Theorem (2,1) implies the Theorem (1,1) 5. The integers ej 6. The numbers g, gT, g", p,
Applications 1. The theorems of Roth and Ridout 2. The continued fraction of a real algebraic number 3. The powers of a rational number 4. The equation pto + Q&) + R
xi 117 120 122 123 125 127 130 131 133 133 134 135 137 140 143 145 147 147 150 150 155 158 161
Appendix A. Another proof of a lemma by Schneider
163
Appendix B. A theorem by M. Cugiani Appendix C. The Approximation Theorems over Algebraic Number Fields
169 181
PARTI p-ADIC AND g-ADIC NUMBERS, AND THEIR APPROXIMATIONS The theory of Diophantine approximations arose from that of indeterminate or Diophantine equations. This accounts for the name; no work by Diophant on Diophantine approximations is known. A characteristic problem of the theory demands finding fractions 2 of small denominators q > 0 that are near to a given real number a . One can then show in many ways that, if a is irrational, there are infinitely many such fractions satisfying
but that this is false if a is rational. In the opposite directions one can show, on one hand, that there are numbers a such that always
where c > 0 is a constant, and, on the other hand, that there also exist numbers a for which the inequality 0< la-
has infinitely many solutions however large the exponent o> > 0 is taken. We shall later study similar problems, but from a more general standpoint. It seems probable that already the Greek mathematicians, in particular Archimedes, had methods for solving the last problem. For Euclid's algorithm for finding the greatest common divisor of two integers is closely connected with continued fractions, and such continued fractions give perhaps the simplest solution. The general theory of Diophantine approximations may be said to have started with Lejeune-Dirichlet who introduced the Schubfach-Prinzip as a very powerful tool, particularly for the study of linear problems. More recently Minkowski founded the Geometry of Numbers which leads to even better results1). Other distinguished contributors to the theory were Jacobi, Liouville, Hermite, Kronecker, Thue, and Weyl, as well as several living mathematicians. It is best not to restrict the theory to the study of real and complex numbers, but to allow other kinds of quantities and in particular the p-adic and g-adic numbers of K. Hensel. As the reader may not have studied such numbers previously I shall begin this course with their construction by means of valuation theory. It will be assumed that the reader is acquainted with the elements of modern algebra, say with as much as is contained in the first five chapters of the well-known book by van der Waerden. I shall also assume a slight 1. We shall not apply the geometry of numbers; it is to form the subject and method of a continuation of these lectures.
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LECTURES ON DIOPHANTINE APPROXIMATIONS
knowledge of analysis and of the more elementary parts of the theory of numbers. However, more difficult or less well-known results from these disciplines will be proved when they ar,e needed. BOOKS FOR FURTHER READING H. Minkowski:
Diophantische Approximationen, Leipzig 1907
J. F. Koksma:
Diophantische Approximationen, Ergebn. der Math. IV, 4, Berlin 1937
E. Lutz:
Sur les approximations diophantiennes line*aires P-adiques, Actuality's scient. et ind. No. 1224, Paris 1955
W. J. LeVeque:
Topics in number theory, 2 vols., Reading 1956
J. W. S. Cassels:
An introduction to Diophantine approximation, Cambridge Tract No. 45, 1957
Chapter 1 VALUATIONS AND PSEUDO-VALUATIONS It is shown in abstract algebra that there are only the following two distinct types of simple extensions of a field K. A simple transcendental extension of K is obtained by adjoining an indeterminate x to K and forming the field K(x) of all rational expressions in x with coefficients in K. Apart from isomorphisms there is only one such extension of K. Next there are the simple algebraic extensions of K of which there may be many. These may be obtained as follows. Denote again by x an indeterminate, further by K [x] the ring of all polynomials in x with coefficients in K, and by f(x) an element of K[x] which is monic (i.e. has highest coefficient 1) and irreducible over K. The polynomials divisible by f(x) form a prime ideal p in K[x]. Divide the elements of K[x] into residue classes modulo p by putting two elements into the same class if their difference is in p. These residue classes form together the residue class ring K[X]/JI which, in fact, turns out to be a field. Furthermore, the residue class, \ say, that contains the polynomial x, satisfies the equation f(£) = 0. In this way K has been extended to a field K[x]/p = Kg) in which the equation f (£) = 0 has at least one root (; . Apart from isomorphisms there is again only one such extension; but different monic irreducible polynomials f(x) will generate different simple algebraic extensions. The construction of both extension fields K(x) and K(£) does not require that K was already imbedded in a larger field, and it uses only algebraic processes. More important for the theory of Diophantine approximations is a non-algebraic method of field extension that is based on ideas from topology. This non-algebraic method is applied already in elementary analysis where it serves to extend the field F of the rational numbers to the larger field P of the real numbers. Of the different variants of this method we select the one which has the advantage of easy generalization. Define a real number a as the limit a = lim am m—•* oo of a convergent sequence {am} = {ai > aa, as,...} of rational numbers; here the sequence is said to be convergent or a fundamental or Cauchy sequence if lim |a m -a n l=0. m—>°° n—*°° Further two fundamental sequences {am} and {bm} have the same limit if and only if lim lam-bm 1 = 0, m—*«> and the special sequence {a, a, a,...} has the rational limit a. 3
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LECTURES ON DIOPHANTINE APPROXIMATIONS
In this manner the rational field F becomes inbedded in the real field P. We may now study subfields of P, and in particular simple extensions F (a) of F where a is any chosen element of P. This construction of a simple extension of F naturally is completely different from that by the abstract algebraic method. It becomes now an interesting problem to decide whether T(a) is a transcendental or algebraic extension of F, or, as we say, whether a is a transcendental or an algebraic real number. This problem will be studied in several of the later chapters. 1. Valuations and pseudo-valuations. The construction Just mentioned of the real numbers depends essentially on the fact that the function |a| is a valuation of F. Here a valuation w(a) of an arbitrary field K denotes any real-valued function of the elements a of K which has the following properties,
(1): (2): (3):
w(0) = 0, but w(a) > 0 if a 4 0; w(ab) = w(a)w(b) (product equation); w(a=Fb) < w(a)+w(b) (triangle inequality). If w(a) has the properties (1) and (3), but instead of (2) satisfies the weaker relation (2 f ):
w(ab) ^ w(a)w(b)
(product inequality),
then w(a) is said to be & pseudo-valuation of K. It is clear that every valuation is also a pseudo-valuation; but the converse need not be true. From these definitions the following properties follow easily. When w(a) is a valuation:
w(-a) = w(a),
W(T 1) = 1,
w(a) - w(b) ^ w(a-b) ^ w(a) + w(b), w(ft aj) = n wfa,,), V=l / *=1 "
|w(a) - w(b) I * w(a-b), wf 2 a*} * E \v=i I v=
When w(a) is a pseudo-valuation:
W(T 1) £ 1,
w(-a) = w(a),
w(a) - w(b) < w(a-b) < w(a) + w(b), w(v=l S aj\ < v=l n w(a^), \ I
|w(a) - w(b) I < w(a-b), w ( 2 sty) < Z \i;=l / i;=
Here n may be any positive integer. Each field has at least one valuation, viz. the trivial valuation defined by
0 if a = 0,
w0 (a) =
1 if a 4 0.
VALUATIONS AND PSEUDO-VALUATIONS
E
For the trivial valuation the triangle inequality (3) holds in the strengthened form Wo(aTb) ^ max{wo(a), Wo(b)]. This we express by saying that wo(a) is Non-Archimedean. More generally, any valuation or pseudo-valuation w(a) of K is called Non-Archimedean if (3') : w(aTb) ^ max[w(a), w(b)] ; but w(a) is said to be Archimedean if this inequality is not satisfied for all a and b. Thus the valuation |a| of F is Archimedean. 2. The p-adic valuations of r. In addition to |a| and w0(a), the rational field F possesses yet infinitely many other valuations. For let p be any one of the infinitely many primes 2,3,5, ..... Denote by |a|p the function on F defined as follows, (i): lolp-0; (ii): if a 40 is any other element of F, let n be the unique integer such than p-n a = - where both integers r and s are prime to p; then put s It is not difficult to prove that | alp is a valuation of F, and that it is NonArchimedean. For | a |p has the property (1); and if at least one of a and b vanishes, then properties (2) and (31) are trivially satisfied. If, however, both a 4 0 and b 4 0, let v be the integer such that p~v b =ffB. where p and or are prime to p. Now (2) follows from
since rp and SCT are prime to p. Assume further that, say n *&v. Then p-n(aTb) =j*L±jp!o
where sa is prime to p, but ra T pI/"nsp may still be divisible by a positive power of p, so that We call |a|p the p-adic valuation of F, and also say that |a|p is the p-adic value o/a. If q is an arbitrary prime, evidently
hip- -1
if
q4p.
This means that p-adic valuations belonging to different primes p are distinct. Therefore F possesses an infinite set of distinct valuations, w0(a), |a|, |a|p for all primes p.
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LECTURES ON DIOPHANTINE APPROXIMATIONS
These valuations are related to one another by the FUNDAMENTAL IDENTITY: |a| U |a|p = w0(a) if ae F, P * a formula which is equivalent to the fundamental theorem of arithmetic on the unique factorisation of integers. The product H in this identity runs over all primes; note that for a ^ 0 all but finitely many of the factors la L are equal tol. Of particular interest is the case when a is an integer distinct from zero when evidently Wo(a) = 1, lal^l, |a|p < 1 for all primes p. Denote by pi, p2,...., pr an arbitrary set of finitely many distinct primes. The fundamental identity leads then immediately to the FUNDAMENTAL INEQUALITY:
|a| H |a|p. *1.
This inequality is basic for our whole theory. Remark: It is shown in valuation theory1) how valuations of a field K can be continued into simple algebraic extensions of K. If K is the rational field, these extensions become finite algebraic number fields A. One finds that, apart from the trivial valuation, A possesses finitely many Archimedean valuations and an enumerably-infinite set of Non-Archimedean valuations. These valuations satisfy again both a fundamental identity and a fundamental inequality very similar to those for F. It is therefore possible to develop a theory of Diophantine approximations over A which is completely analogous to that over F which is treated in these lectures. Very little new is required for the more general theory; but formulae naturally become rather more involved. For this reason I shall not deal with the more general theory except in Appendix C. 3. A further example.
There is yet another class of fields for which a theory of Diophantine approximations may be developed. This are the simple transcendental extensions K = S(x) of an arbitrary field S. Of main interest for the theory are those valuations of K that become identical with the trivial valuation when a lies in the ground field S. The following construction produces infinitely many of them. Denote again by S[x] the ring of polynomials in x with coefficients in S; let further p be any "prime", i.e., a monic irreducible polynomial in S[x], Also let 0 be a real constant such that 0 < 6 < 1. If r 4 0 is an arbitrary element of £[x], we shall write deg r for the degree of r and ordp r for the order of r at p, i.e., the largest integer g such that pS is a factor of r. Every element a =1= 0 of K may now be written in a unique way as a quotient a = § of two polynomials r and s in S[x] that are relatively prime and where s is monic and not the zero polynomial. In terms of r and s put 1. See e.g. the 10th chapter of Modern Algebra by van der Waerden.
VALUATIONS AND PSEUDO-VALUATIONS
0 0degs-degr
if a = 0, if a M>
0 if a = 0, 00rd p r-ord p s tf a ^Q
There is no difficulty in proving that these two functions of a are Non-Archimedean valuations of K and that wo(a), ||a||, I (a I |p for all "primes" p form a system of distinct valuations. One may adapt Euclid's method for proving that there are infinitely many rational "primes" and so prove2 that this system has infinitely many elements. Since the polynomials in S[x] satisfy the law of unique factorisation.into "primes", there is again a fundamental identity. It takes in this case the form where the product II extends over all "primes" p. There is also a fundamental inequality: if pi , p2 ,..., pr are finitely many distinct "primes" and a ^ 0 is an element of S[x], then IklljJ IkllJ"* * 1 .
These two properties allow to develop a theory of Diophantine approximations over K = S(x) which proves to be very similar to that for T treated in these lectures. Although we shall not deal with this theory, the valuations of K will occasionally be used by way of example. 4. Valuations and pseudo- valuations derived from given ones.
Let K be any field and w(a) any valuation or pseudo-valuation of K. If X is any constant such that 0< \ < 1, put w*(a) = w(a)\ Then w*(a) is likewise a valuation or pseudo- valuationf of K, respectively. For that w*(a) has the properties (1) and (2) or (2 ) is obvious, but we still have to prove that also the triangle inequality (3) is satisfied. Now one shows easily that if x and y are non-negative real numbers, then3) Hence from the properties (1) and (3) of w(a), 2. If pi, pai. . ., pr are finitely many distinct "primes" in S [x], pipa . . .pr +1 is divisible by a ''prime" distinct from the given ones. 3. For fixed x ^ 0 and variable y > 0 the function f (y) = (x + y)^ - y^ has the derivative --p* = \{(x +y)^~:Lyk-1}
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LECTURES ON DIOPHANTINE APPROXIMATIONS
w*(a+b) = w(a+b)x < {w(a) + w(b)}x ^ w(a)x + w(b)x = w*(a) + w*(b), as asserted. If w(a) is Non- Archimedean, this proof becomes superfluous. It is, in fact, now obvious from (3 f ) that, if X is any positive constant, w*(a) = w(a)x is likewise a Non- Archimedean valuation or pseudo- valuation, respectively. Next let Wi (a) ,..., Wr(a), where r ^ 2, be finitely many valuations or pseudo-valuations of K. Then the maximum , 3=1,2,. ,.,r
is again a pseudo- valuation of K. For ws(a) trivially has the property (1). Next, ws(ab)= max wj(ab)< max Wi(a)wi(b)^ max wi(a) . max Wi(b) = j=l,2,...,r j=l,2,...,r J j=l,2,...,r j=l,2,...,r =ws(a)ws(b), and so w^(a) has the property (2T). Finally, ws(a=Fb)= max wj(aTb)< max (wj(a)+wj(b)}< j=l,2,...r j=l,2,...,r ax Wj(a)+. max Wj(b)=w s(a)+wy(b), ,2,...,r J j=l,2,...,r J * ^ which shows that ws(a) also satisfies the triangle inequality (3). We say that w^(a) is the sum of Wi (a),..., wr(a), and we call Wi (a),..., wr(a) the terms of ws(a). If all these terms are Non -Archimedean, it is again easy to see that their sum is likewise Non- Archimedean; for now = max max max{wj(a),Wjfti)} = j=l,2,...,r =max
max Wf(a), max wi(b)7=max{ws(a),W2;(b)}. j=l,2,...,r j=l,2,...,r J
Of particular interest for us is the case when all terms Wi (a),..., wr(a) of ws(a) are valuations. This does not imply that w^(a) is necessarily also a valuation. For instance, in the trivial example K = r,
ws(a) = max(|a|, |a|¥),
we have ws(f-) = (*•)*, ws(2) = 2, ws(f 2) = 1 ± (|-)^-2. But although ws(a) need not be a valuation, it does have two simple properties that a general pseudo-valuation need not have: (I): If n is a positive integer, then for all a in K, ws(an) = ws(a)n. (II): // g =(= 0 is an element of K such that ... = Wr(g) = WS(g),
VALUATIONS AND PS EUDO- VALUATIONS and if a is an arbitrary element of K, then ws(ag) = ws(a)ws(g) and more generally, wj^ag11) = ws(a)ws(g)n for all integers n. These properties follow immediately from the product equation (2) for the terms Wj(a) of w^(a). By way of example, the rational field F has the pseudo- valuations and X
1
max( l a l , |I aa,l^|p ,..., lal**lpl, l&r }/ UldA.V|d|
where pi,..., Pr are finitely many distinct primes and X, Xi,..., Xr are real constants satisfying 0<X * 1, X i > 0 , ... ,X r >0. These pseudo-valuations of r have then the properties (I) and (n). The constructions of this section are by no means the only ones that allow to derive new valuations or pseudo-valuations from given ones, but they suffice for the purpose of these lectures. For other operations the reader is referred to a joint paper by P. Cohn and myself4). 5. Bounded sequences, fundamental sequences, and null sequences.
In the next few sections we shall now generalise the method of the introduction for defining real numbers as limits of convergent sequences of rational numbers. Let K be an arbitrary field with the valuation or pseudo-valuation w(a), and let be an arbitrary infinite sequence of elements of K. Such a sequence is said to be a bounded sequence with respect to w(a) if there exist two positive numbers p and M such that
w(am) < M for all m ^ p; it is said to be a fundamental sequence with respect to w(a) if, given any e > 0, there is a positive number q(e) such that w(am-an) < e for all m ^ q(e) and all n ^ q(e); and it is said to be a null sequence with respect to w(a) if, given any e > 0, there is a positive number r(e) such that w(am) < e for all m ^r(e).
4. Nieuw Archief voor Wiskunde (3), 1 (1953), 161-198.
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LECTURES ON DIOPHANTINE APPROXIMATIONS
It is obvious that z/ {am} is a bounded, or a fundamental, or a null sequence, then {-ai» -a2, -as,—} is a sequence of the same kind. The following lemmas also are easy consequences of the definitions. (a): Every fundamental sequence is bounded. For let p be an integer not less than q(l), and let m ^ p. Then w(am) = w{ap + (am-ap)} < w(ap) + w(am-ap) < w(ap) + 1, = M say. (b): Every null sequence is a fundamental sequence. For let m ^ r(f-e) and n ^ r(|-e). Then w(am) < re, w(an) < f-e, w(am-an) ^ w(am) + w(an) < |-e + f-e = e. (c): If {am} is a fundamental sequence, but not a null sequence, there exist two positive numbers s and N such that w(am) > N for all m ^ s. For assume that {am} is a fundamental sequence, but that there are no such numbers s and N. Then, however small e > 0 is chosen, there exist arbitrarily large suffixes n such that w(an) < je. There is then also a suffix n of this kind satisfying n ^q(f-e), and now also w(am-an) < |-e for all m^q(f-e). Hence, for all m ^q(|-e), w(am) = w{an+(am-an)} * w(an) + w(am-an)< f-e + ^-€ = e, and so {am} is a null sequence. In the next two lemmas we denote by p', M1, q f (e), and r f (e) the numbers corresponding to p, M, q(e), and r(e), respectively, that belong to the second sequence {bm}. (d): If {am} and {bm} are fundamental sequences, so are {am+bmK {am-bm}i and {ambm}. For first, w{(amTbm)-(an:Fbn)}=w{(am-an)=F(bm-bn)hw(am-an)+w(bm-bn)
provided that
VALUATIONS AND PSEUDO-VALUATIONS ^
m > max
(e):
I
I € \
t/ € \
11
fl
q[-^n= 1, qf[ «^ 1 , p,pf and
If {am} and {bm} fl^e ^^^ sequences, so are {am+bm} «^ {am-bm} If {am} te « null sequence, and {bm} te bounded, then {ambm} te fl null sequence. sequence. The first assertion holds because w(am) + w(b m )< g c + 2C = c is
m
*mH3\2c) >r \2e) J
and the second one because w(ambm) * w(am) w(bm) < ^ M1 = e if m > max rh^ j , p1
.
6. The ring {Kjw and the ideal j> . Denote by {K}W the set of all fundamental sequences of K with respect to the valuation or pseudo- valuation w(a). On account of (d) we may then define for the elements of {K}w operations of addition, subtraction and multiplication by the formulae, {amMbm} = {am+bm}» {am}- {bm} = {am-bm}i {amHbm} = {ambm} It is easily verified that, with respect to these operations, {K}w satisfies the commutative, associative, and distributive laws of addition and multiplication, and that subtraction is the inverse operation to addition; further {0} = {0,0,0,...} and {1} = {1,1,1,...} are the zero and unit elements of {K}w. Since, e.g. {1,0,0,0,...} {0,1,0,0,...} ={0}, {K}w is then a commutative ring with unit element, but is neither a field nor even a domain of integrity on account of these zero divisors. Let ft denote the subset of {K}w consisting of all null sequences. (f): The set p is an ideal of {K}w. // w(a) is a valuation, then jt is a prime ideal of Ww. From (e), if {am} and {bm} are elements of p, so are {am} + {bm} and {am} - {bm}; if further {am} belongs to f and {bm} to {K}m, then {am}{bm} is an element of f. Hence jt is an ideal. Next assume that w(a) is a valuation. It suffices to show that if {am} and {bm} belong to {K}w, but not to p, then their product likewise is not an element of p . By fc), the hypothesis implies that there are four positive numbers s, N, s' , and N* such that w(am) > N if m ^s and w(bm) > N» if m ^s1. Hence w(ambm) = w(am)w(bm) > NN1 if m ^ max(s,sf), and hence {am} {bm} is not a null sequence. When w(a) is not a valuation but only a pseudo-valuation, this proof is not valid, and then p need not be a prime ideal.
12
LECTURES ON DIOPHANTINE APPROXIMATIONS
7. The residue class ring Kw. If the difference {am} - {bm} = {am-bm} of two fundamental sequences {am} and {bm} lies in p , i.e., is a null sequence, then the two sequences are said to be congruent modulo p , and we write {am} = {bm} (mod *t), or simply {am} ={bm}This is in agreement with the usual notation of congruence modulo an ideal. It is also well known that such congruence is an equivalence relation; i.e., in the present case the following three laws hold: {am} s{am}j if {am} s {bm}, then {bm} s {am}; if
{am} s {bm} and {bm} a {cm}, then {am} a {cm}-
For {am} - {am} = {0} is a null sequence; if {am} - {bm} is a null sequence, so is {bm} - {am} = -({am} - {bm}) > and if {am} - {bm} and {bm} - {cm} are null sequences, so is
{am} - {cm} = ({am} - {bm}) + ({bm} - {cm}) . On account of this property, we may subdivide the elements of {K}w into classes by putting into the same class all fundamental sequences that are congruent modulo ft to one given fundamental sequence. Denote by Kw the set of all such residue classes modulo ft; in the notation of algebra, Let a and 0 be two elements of Kw, and let, say, {am} and {afm} be any two fundamental sequences in the residue class a and {bm} and {bm} any two fundamental sequences in the residue class |3. Hence both and {bm-b'm} are null sequences. It follows then that also {(anrFbm)- (am*bm)} = {am-a'm} T {bm-bm}
and {ambm-ainbm} = {am-am} {bm} + {am} {bm - bm} are null sequences, and that therefore {am}-{bm}={am}-{bm}, {am}{bm} We have thus proved that the residue classes, a+]8, a-)3, and a/3 say, that contain the fundamental s equences {am+bm}> {am-bm}> and {ambm}> respectively, remain unchanged if {am} is replaced by any congruent sequence {am}, and {bm} is replaced by any congruent sequence {bm}. Hence Kw admits the operations of addition, subtraction, and multiplication and so is a ring. It is also obvious, from the definitions of these operations, that addition and multiplication in Kw are commutative, associative, and distributive, and that subtraction is the inverse operation to addition. If a is any element of K, the ring Kw contains the special residue class, (a) say, that is defined by the fundamental sequence {a} = {a, a, a,...}. In particular, (0) is the zero element of Kw and is identical with the set of all null sequences, and (1) is the unit element of Kw. Since, evidently,
VALUATIONS AND PSEUDO- VALUATIONS
13
(a) + (b) = (a+b), (a)- (b) = (a-b), (a)(b) = (ab) , the elements (a) of Kw form a ring that is isomorphic to K. In fact, they form afield isomorphic to K, (K) say. For (a) is distinct from (0) if and only if a k 0, and then (a) has the inverse (a"1 ) because (a)(a"1) = (1). From now on we shall not distinguish between the element (a) of (K) and the corresponding element a of K, thus shall identify (K) with the original field K. It may then be said that the ring KW of all residue classes modulo p contains K as a subfield. We call KW the completion of K with respect to w(a). 8. The completion of a field with respect to a valuation.
Of particular interest is the case when w(a) is a valuation. (g): If w(a) is a valuation of K, then the completion Kw is afield. It suffices to show that if a is any element of Kw distinct from zero, there exists a second element of KW, of1 say, such that a a'1 = 1. Let {am} be an arbitrary fundamental sequence in the residue class a ; since of 4 0, this sequence is not a null sequence. There are then by (c) two positive numbers s and N such that w(am) > N and hence am 4 0 if m ^ s. Put
am = 0 if 1 ^ m <s,
am = — if m > s. am
The new sequence {aj^} is likewise a fundamental sequence because, for m ^s and n ^ s,
»«*-"«> - « -
-<
•<>»-*>•
and hence w(am-a$) < e if m ^max{q(N2e),s), n ^max{q(N2e),s}. Denote by a"1 the residue class of {am}. Then a a"1 = 1, since {am} {am} = {amamK where amam = 1 if m ^ s. When w(a) is only a pseudo-valuation, K^ in general will not be a field, but may contain divisors of zero. One such case will soon be discussed. 9. The limit notation.
Let again K be a field, w(a) a valuation or pseudo- valuation of K, and Kw the completion of K with respect to w(a). It is convenient, and in agreement with the usual convention for the real field, to adopt the following notation. If {am} is any fundamental sequence, and if a is its residue class in KW, then we say that a is the limit of am with respect to w(a) as m tends to infinity, and we write
14
LECTURES ON DIOPHANTINE APPROXIMATIONS
a = mlim —»oo am (w). From the definition of a, this limit is naturally unique. For the fundamental sequence of the special form {a} = {a, a, a,...} we have a= lim a (w) m—>°° since this sequence lies in the residue class (a) which we have identified with the element a of K. The definition of the operations in Kw immediately implies that, if ]8 = lim bm (w) m-* <» is a second limit, then of + ]8= lim (am+km)(w), « - | 3 = lim (am-bm)(w), aj8 = lim (ambm)(w). m—* °o m—* °o m—»°° Let, in particular, w(a) be a valuation, and assume that ]3 4 0 and that all bm are distinct from zero. It follows then from (g) that IS'1 = lim * (w). m—> °° "iXL Hence in this case also ? = a fl"1 = lim 10. The continuation of w(a) onto Kw.
Let
a = lim am(w) . m—»°o From the definition of a fundamental sequence with respect to w(a), lw(am) - w(an) | ^ w(am-an) < c if m ^ q(e) and n ^ q(e) . Hence {w(am)} is a convergent sequence of real numbers, and the real limit lim w(am), = W(a) say, m—»°° exists. (h): W(a) depends only on a, and not on the fundamental sequence as the limit of which a is defined. For let also a = lim am(w). m—»°° Then {am-afin} is a null sequence and therefore lim w(am-am) = 0. m—»°° It follows then that
VALUATIONS AND PS EUDO- VALUATIONS
15
0^1 lim w(a m )- lim w(a m )| = lim |w(am)-w(am)|< lim w(am-am) = 0, m->«> m-»°o m-**> m-+°o whence lim w(am) = lim w(am). m— » °o (i): W(a) = w(a) when a is any element of K. For now a = (a) is the limit of the fundamental sequence {a} and hence W(a) = lim w(a) = w(a). m— >°° (j):
W(a ) is a pseudo-valuation of Kw. It is a valuation if w(a) is a valuation; it is Archimedean if w(a) is, and it is Non-Archimedean if w(a) is. First, the definition of W(a) implies that this function vanishes only if a = 0 and is otherwise positive. Secondly, W(aj3)= lim w(ambm)< lim w(am)w(bm)= lim w(am) lim m—»°o m-»«> m->°° m—»°° with equality if w(a) is a valuation. Third, lim w(am=Fbm)^ lim (w(am)+w(bm)) = limw(a m )+ lim w(bm) = <= W(a) + W(/3) .
When w(a) is Non-Archimedean, we have instead W(ofTj3) = lim w(amTbm) ^ Jim_ max{w(am),w(bm)} : m—*°° = max{ lim> w(am), lim w(bm)} = max(W(a), m— °° m—*°° and it is trivial that if W(a) is Non-Archimedean, so is w(a). The valuation or pseudo-valuation W(a) is called the continuation of w(a) onto the completion Kw. Since Wfe) and w(a) are identical on K, and w(a) has so far only been defined on K, we shall in future write w(a) for W(a) whether a is an element of K or of Kw. 11. The elements of K lie dense in Kw.
In real analysis, the limit a of a convergent sequence {am} of rational numbers satisfies the limit relation lim I a - an I = 0. m—*°o We show now that an analogous relation holds also for K and Kw. Let Q? = lim am (w) . m—*°o
16
LECTURES ON DIOPHANTINE APPROXIMATIONS
Since also an = lim an(w) m—*°°
(n = 1,2,3,...),
tf-an = lim (am-an)(w) m—>°°
(n = 1,2,3,...)
w(a-a n ) = lim w(am-an) m—»°°
(n = 1,2,3,...).
it follows that
and hence
Now, by the hypothesis, w(am-an) < e if m ^ q(e) and n ^ q(e), and therefore 0 < w(a - an) = lim (am-an) <eit n & q(e). m—»°° Since e > 0 may be arbitrarily small, this means that lim w(a - an) = 0. n—*°° Thus the elements of Kw may be approximated arbitrarily closely with respect to w(o?) by the elements of K. 12. Fundamental sequences in Kw.
The construction in §§ 5-9 of Kw, the completion of K with respect to w(a), is independent of the hypothesis that K is afield and remains valid when K is any commutative ring. In particular, even under this more general assumption the following properties still remain true: Null sequences are fundamental sequences; sum and difference of two fundamental sequences are fundamental sequences; and sum and difference of two null sequences are null sequences. Let us apply this remark to the ring Kw and its valuation or pseudovaluation w(a). We may form "new" fundamental sequences and null sequences consisting of elements of KW instead of K, and by means of these construct the completion, (Kw)w say, of Kw with respect to w(a). However, as will now be proved, the "new" fundamental sequences have already limits in Kw itself, and so this complicated procedure is unnecessary. (k): Let {am} = {ai, a2, a3,.«-} be ajundamental sequence of Kw with respect to w(a). There exists an element a. of Kw such that lim w(a-am) = 0. m—»oo By what was proved in § 11, we can find for each am an element am of K such that w(«m-am) is arbitrarily small, say w(am-am) < -5 The sequence
(m = 1 2 3
' ' >—)•
VALUATIONS AND PSEUDO-VALUATIONS
17
is therefore a "new" null sequence and hence also a "new" fundamental sequence. The difference of the two fundamental sequences, {ami = {^m} - {«m"am} is therefore likewise a "new" fundamental sequence. But since its elements am belong to K, and because w(ff) is the continuation of w(a), it follows that {am} is a fundamental sequence of K with respect to w(a). Let
a = lim am(w) m—»«> be its limit in Kw. Then, by § 11, lim w(ff-a m ) = 0, m—>°° so that {of-am} is a "new" null sequence. But then the difference of the two null sequences {a-flm} = fa-am} - {«m-am} is likewise a "new" null sequence, whence the assertion. This result shows that every "new" fundamental sequence {am} differs only by a "new" null sequence {a-flm} from a "new" fundamental sequence of the special form {a} = {a, a, a,...} where aeK. Thus the completion (Kw)w of KW is isomorphic to, and may be identified with, Kw. We may also write a = lim om(w), m—>«> just as is done in real analysis. Then these "new" limits satisfy the same rules as did the limits of fundamental sequences {am} of K. It is for these reasons that KW was called the completion of K with respect to the valuation or pseudo-valuation w(a). 13. Equivalence of valuations and pseudo-valuations. Let again K be a field, and let w(a) and w' (a) be two distinct valuations or pseudo-valuations of K. We say that w(a) and w! (a) are equivalent and write w(a)~w'(a) if every null sequence of K with respect to w(a) is also a null sequence with respect to wf (a), and vice versa. This definition implies that also every fundamental sequence of K with respect to w(a) is a fundamental sequence with respect to w* (a), and vice versa. For assume jam} were a fundamental sequence with respect to w(a), but not with respect to w* (a). There would then evidently exist an infinite sequence of pairs of suffixes (mi,ni), (m2,n2), (m3,n3),... such that
^im mk = lim n^ = «>, and that further
18
LECTURES ON DIOPHANTINE APPROXIMATIONS
is a null sequence with respect to w(a), but not with respect to w'(a)» against the hypothesis. It follows that equivalent valuations or pseudo-valuations w(a) and wf(a) generate identical and not only isomorphic completions of K: KW=KW< . From the standpoint of valuation theory alone there would then be no need to distinguish between such equivalent valuations or pseudo-valuations. Two equivalent valuations w(a) and wT(a) can be proved to be connected by an identity w'(a) = w(a)x where X is a positive constant. Such a simple relation need not hold between two equivalent pseudo-valuations. For instance, in the case of the rational field T,
|a| andmax(|a|, |a|*) are equivalent. This trivial example also shows that a valuation may well be equivalent to a pseudo- valuation. 14. The valuations and pseudo-valuations of F. The relation w(a)~w'(a) is an equivalence relation in the algebraic sense. We may therefore subdivide all valuations and pseudo-valuations of a given field K into equivalence classes, and then the problem arises of determining all distinct equivalence classes. A. Ostrowski was the first to determine a full system of nonequivalent valuations of any finite algebraic number field5), and I did the same for the classes of non- equivalent pseudo-valuations of such a field6 ). For our purpose the most important case is that of the rational field F. Then the result is that every valuation of T is equivalent to one of the valuations Wo(a), |a|, and |a|p where p runs over all primes p = 2, 3, 5,...; and every pseudo-valuation of F which is not already equivalent to one of these valuations must be equivalent to a pseudo-valuation of the form or wa(a) = max(| a x Wl (a) = max( |a ) , . . . , |a| Here PI,..., Pr are finitely many distinct primes, with r ^ 2 in the case of Wi(a) and r **1 in that of w2(a); and X, Xi,..., Xr are constants satisfying 0 < X *1, Xi > 0,..., Xr > 0. An alteration of these constants has only the effect of replacing Wi(a) or w2(a) by an equivalent pseudo- valuation, and so, from the standpoint of valuation theory, it would suffice to put 5. Acta math. 41 (1919), 271-284. 6. Acta math. 67 (1936), 51-80.
VALUATIONS AND PSEUDO-VALUATIONS
19
X = Xi = ... = Xj» = 1.
However, for the later applications to Diophantine approximations, a different choice of these constants is of advantage. For let g ** 2 be an arbitrary integer, and let
g = Pie'...pfr be its factorisation into a product of powers of different primes Pi ,•••> Pr with exponents ei ,..., er that are positive integers. Fix now Xi,..., Xr such that | g | ^ = . ... . . = l g £ r - J , i.e., take PI Pr Jog_g_ 1
logg e r logpr '
and put log g
log g
|a|g=max(|a|£ logpl ,..., lalj10"*)
and log g logpl
|a|g* = max(|a|, |a|^
log g ',..., |a|glogpr ) = max(|a|,|a|g).
We call |a|g,and |a|g* the g-adic and the g*-adic pseudo-valuations, respectively, and also speak of the g-adic and g*-adic values of a. The definition of |a|g implies that for all a in F and all integers n. This is easily verified and is also contained in the property (II) of § 4. If g' ^ 2 is a second integer, it is obvious that |a|g and |a|gf are equivalent if and only if g and gf have the same prime factors Pi,..., Pr and differ only in their exponents; and just the same holds for |a|g* and |a|g»*. Furthermore,
|a|g = |a|p if g is a positive integral power of the single prime p. In these lectures, P will denote the real field, i.e., the completiorkof P with respect to |a|; and similarly Pp. Pg, and Pg* will stand for the completions of F with respect to I alp, |a[g, and |a|g*, respectively. Then Pp, Pg, and Pg* are the field ofp-adic numbers, and the rings of g-adic and g*-adic numbers, respectively. This field and these two rings were introduced by K. Hensel7 ) in 1892 and have proved of fundamental importance in many branches of mathematics. * We shall study the elements of Pp, Pg, and Pg* in detail in the next 7. HensePs little book Zahlentheorie (Berlin 1913), which gives an elementary introduction to the theory of p-adic and g-adic numbers, may be particularly recommanded on account of its many examples of actual computations with such numbers.
20
LECTURES ON DIOPHANTINE APPROXIMATIONS
chapter; and they will form both the object and a tool in most of the later work. It is clear that, while |a| and |a|g* are Archimedean, I alp and |a|g % are Non- Archimedean. This was proved for |a|p in §4 and so follows for |a|g from the definition. It is sometimes convenient to define |a|g also in the excluded case when g = 1, by putting |a|i = w0(a). 15. Independent pseudo-valuations. The g-adic ring Pg and the g*-adic ring Pg* can be decomposed into finitely many field P and Pp, as was already proved by Hens el7). We shall prove this decomposition as a special case of a more general theorem on ps eudo- valuations . Denote by K a field, by wi (a),..., wr(a) finitely many valuations or pseudo- valuations of K, and by 1 if h = k, 0 if h 4 k the well-known Kronecker symbol. Then wi(a),..., wr(a) are said to be independent if there exists for each suffix h = 1, 2,..., r an infinite sequence
{da(hU(h);dlh),...} in K such that lim Wk(dj?- 6hk) = 0
(k = l,2,...,r),
ejik = lim djj (wk) m —>°°
(k = l,2,...,r).
or, what is the same, that
From this definition, it is immediately clear that, if wi(a)~ wi(a),..., w' r (a)~ wr(a), then also wi(a),..., wr(a) are independent. By way of example, let us consider the rational field F. Here the following result holds. (1):
y ?!»•••> Pr are finitely many distinct primes, then the valuations lal, la|pl,..., lalpr and hence also the valuations |a|pl,..., |a|pr are independent. First, the sequence m \v>\ l m >, whprp where drfm -- (plpa ...— Pr)
VALUATIONS AND PSEUDO -VALUATIONS
21
is easily seen to have the limits 1 with respect to I a I and 0 with respect to |a|pl,..., |a|pr, respectively. Secondly, select for each suffix h = 1, 2, ...,r a positive integer a^ such that Phh > PiPa-.PrThen it is not difficult to verify that the sequence />)} where d fr) (piP2...Pr)m idm J , where dm/ - ( Pl p 2 ...p r )m +p a h m » has the limit 1 with respect to lalp^, but is a null sequence with respect to I a |, as well as with respect to all I a |pk where k 4 h. Kgi^2,..., gr ^ 2 are finitely many integers which are relatively prime in pairs, one shows by a similar proof that also |a|, |a|gl,..., |a|gr and hence also |a|gl,..., |a|gr are independent. 16. The decomposition theorem.
Let again K be a field, and let wi(a) ,..., wr(a), where r ^ 2, be finitely many independent valuations or pseudo- valuations of K. As in § 4, we put ws(a) = max[wi(a),..., w r (a)] and further write wjtya) = max wk(a) (h = l,2,...,r). These functions are likewise valuations or pseudo-valuations of K, and w«(a) is, what we call the sum of Wi (a),..., wr(a). The following lemma gives the justification for the term of "independent" valuations or pseudo- valuations. (m): Let ^eK^,..., areKWp be arbitrary elements of the completions of K with respect to Wi (a),..., Wr(a), respectively. Then there exists an infinite sequence {am} in K such that, simultaneously, lim am = a n (wh)
^h = 1 > 2 v> r )«
First, ai,..., ar are defined as the limits lim a } ( w h ) m—»°°
(h =
of certain infinite sequences {aW},..., {a^ty in K. Secondly, by the definition of independence, there also exist r infinite sequences {dm)},..., {dW}
22
LECTURES ON DIOPHANTINE APPROXIMATIONS
in K satisfying )
(h,k = l,2,...,r),
In particular, each such sequence {dmT» is a null sequence with respect to all wk(a) where k ± h, and so it is a null sequence also with respect to
w<2h>(a).
There exists then, for each h, an infinite sequence of strictly increasing suffixes mfci, mh2> mh3>— such that
Thus
and hence, from the definition of wyO (a), (A):
Ihn d^
ajh) = 0 (wk)
(h,k = l,2,...,r; h*k).
On the other hand8),
? = 1 (wh) (h so that (B):
^m d
a!
= Urn d
£m a
= 1. «h = «h(wh)
(h = l,2,...,r).
On combining (A) with (B), it follows that the new sequence {am} where ai= Z dmhiaih) h=l
(1 = 1,2,3,...)
has the required limits a\ ,..., ar with respect to Wi (a),..., wr(a), 'respectively. We finally prove the following decomposition theorem which establishes the connection between the completions KW, KWl,..., KWr of K. (n):
There is a one-to-one correspondence
8. Let-fbuj} be a fundamental sequence with respect to w(a), and let (bm]}i where nii < m2 <ms < . . ., be an infinite subsequence of-fb^. Then {to -bm , b2-b bs-bm » . . .} evidently is a null sequence with respect to w(a), whence lim bm = lim bmi(w) . m—»oo l—> oo
VALUATIONS AND PS EUDO- VALUATIONS
23
between the elements a of Kws and the ordered sets («i ,..., «r) of one element in each of KWl ,..., KWr such that, if a— -(ai,,.., ar)
and /3 —-(ft,..., /3r),
$&en also correspondence is defined by lim am(ws),
a^ = lim a m (wh)
(h = l,2,...,r),
{am} te a sequence in K wfo'cfe is a fundamental sequence with respect to all of wi (a),..., wr(a), and ws(a). • First, we note that it is obvious from the definition of w^(a) that a sequence {am} in K which is a fundamental sequence with respect to each of Wi(a),..., wr(a) is also a fundamental sequence with respect to ws(a), and vice versa; similarly, if the sequence is a null sequence with respect to each of Wi(a),..., Wr(a), then it is also one with respect to ws(a), and vice versa. Now, by lemma (m), the arbitrary elements aj.eKWl ,..., areKWr can be defined as limits an = lim am(wh) m— » °° of the same sequence {am} in K, and then the limit
(h = 1,2,. ..,r)
a= lim am(w;s)
defines an element a of Kw^. Conversely, if a is given as the limit with respect to w^(a) of such a sequence .{am}> then the limits of {am} with respect to wi(a),..., Wr(a) likewise exist and define elements ai,..., ar of KWl,..., KWr, respectively. The relation a -*^(ai ,..., ar) is independent of the special sequence {am} used in the definition of a, ai9..., ar. For if (C): lim am= lim am(wn) (h = l,2,...,r), m—*°° m—»°o then {am-am} is a null sequence with respect to each of wi(a),..., wr(a) and hence also with respect to w^(a); therefore (D):
lim am = Um a m—»«)
Conversely, (D) implies again (C). The formulae for a+0, a-0, and a/3 finally follow at once from the rules for limits proved in § 9. If a-*^(ai,..., ar), oJi,..., ar are called the components of a. All the completions KWl,..., KWr, and KWS are extensions of K, and every element a of K lies simultaneously in each of these r+1 rings or fields. For such and only for such elements the correspondence relation takes
24
LECTURES ON DIOPHANTINE APPROXIMATIONS
the simple form a ~^""^" \a, a,..., a j. In particular, 0-~<0,0,...,0) and 1—(1,1,...,!) are the zero element, and the unit element, of KWs. We note that, since r ^ 2, KWs is not a field because by afi = 0, where a—-(1,0,...,0), |3—-*(0,1,...,0), KWs contains non-trivial zero divisors. This also implies that ws(a) cannot be equivalent to a valuation since then KW would be a field. In the special case when Wi(a),..., wr(a) are valuations, the completions KWl,..., KWr (but, of course, not KWS) are fields. If now a is divisible by 0, with the quotient
a The for us must important case of the decomposition theorem concerns the g-adic and the g*-adic numbers. If g has the distinct prime factors pi,..., Pr, the correspondence relates the g-adic number a to the ordered set of one pi-adic number oti , one pz-adic number a2j etc., and one pr-adic number ar; g*-adic numbers ot*++.(a0,ai,..., ar) in addition have also a real component a0. The components (a0), «i,-«, «r of a or a* are independent of one another and may be any numbers in the corresponding fields (P), Ppl ,..., Ppp. It is thus in general impossible to deduce from properties of one of the components any properties of the other components. 17. Convergent infinite series.
Let again K be an arbitrary field with the valuation or pseudo-valuation w(a). As in real analysis, it is convenient to introduce the notion of a convergent series. The infinite series GO
Z am = ai+a2+as + ..., where am^K for all m, m=l is said to be convergent with respect to w(a) to the sum a if
/ m
\
\ E ak? = {ai,ai+a2,ai+a2+a 3 ,...} lk=l )
VALUATIONS AND PSEUDO-VALUATIONS
25
is a fundamental sequence with respect to w(a) of limit a, and it is otherwise called divergent. From this definition, the series converges if and only if» given any e > 0, there is a positive number q(e) such that
(
m n v £ ak - £ at) = w(an+i + an+2 + ... + a m )< e k=l k=l /
for all integers m, n satisfying m > n ^ q(e). Since w(an+i + an+2 + ... + am) * w(an+l) + w(an+2) + ... + w(am), 00
the series
^
a
m certainly converges if the series of real numbers oo
E w(am) m=l
converges; but the converse need not even be true in real analysis where w(a)=|a|. On taking m=n+l, it is also obvious that Y am cannot be convergent m=l unless lim am = 0 (w); m—>°o
but, just as in real analysis, this condition is not in general sufficient for convergence. There is, however, one important case when it is sufficient, viz. that when w(a) is Non-Archimedean. For now w(an+l + an+2 + ... + am) ^ max[w(an+l),w(an+2),.-, w(am)]. and so, if {am} is & null sequence with respect to w(a), the righthand side is smaller than e f or all m > n In the following chapter these simple remarks on convergent series will be applied to series for p-adic, g-adic, and g*-adic numbers. Final remark: The sketch of valuation theory given in this chapter has been strictly limited to those facts that are to be applied later. For further study of this interesting and important theory the following texts may be referred to: E. Art in, Algebraic numbers and algebraic functions I, Princeton 1951. H. Hasse, Zahlentheorie, Berlin 1949. O. F. G. Schilling, Theory of valuations, Math. Surveys IV, Amer. Math. Soc. 1950. H. Weyl, Algebraic theory of numbers, Princeton 1940.
Chapter 2 THE p-ADIC, g-ADIC, AND g*-ADIC SERIES Historically, K. Hensel was led to his p-adic and g-adic numbers by considerations of analogy to function fields. Let S be the complex number field, x an indeterminate, and K = S (x) a simple transcendental extension of S; let further w(a) be any valuation or pseudo-valuation of K with the property C, i.e., such that w(c) = w0(c) if ceS, where Wo(a) denotes the trivial valuation defined in §1 of Chapter 1. It can be proved that every valuation with the property C must be equivalent to one of the valuations wo(a), llall, ||a||pr introduced in § 3 of Chapter 1; however, now every "prime" p has the special form p=x-c where ceS because S is algebraically closed. One can further show that every pseudo-valuation with the property C either is equivalent to one of these valuations, or it is equivalent to a pseudo-valuation of one of the two forms wi(a) = max(||a||pl,..., ||a||pr) and wa(a) = max(||a||, l|a||pjl,...,||a||pr). Here Pi = x - ci,..., pr = x - cr, where ch =(= % if h 4 k, are finitely many distinct "primes", and we have r ^2 in the case of Wi(a) and r ^ 1 in that of wa(a). The position is thus analogous to that mentioned in § 14 of Chapter 1 for the rational field T , with Hall, ||a|L, w^a), W2(a) corresponding to I a I, I a I p, I a I g, I a I g*, respectively, There is, however, the difference that all these valuations and pseudo-valuations of K are Non-Archimedean. It is not difficult to prove that the completion of K with respect to l l a l l is the field of all formal series
while that of K with respect to I la Up, where p=x-c, is the field of aM formal series cf(x-c)f + cf+i(x-c)f+1 + Cf+2(x-c)f+2 + ... . In both cases f may be any rational integer, and the coefficients cm may be arbitrary elements of S. The convergence of the series follows from the results in § 17 of Chapter 1 because H i l l - e< 1, ||cmll< 1, and ||x-c|L=0 < 1, ||c m |L«l f 26
THE p-ADIC, g-ADIC, AND g*-ADIC SERIES
27
respectively, and hence lim ||c m (M || = 0, lim l|cm(x-c)m||p = 0. \x/ m—»°° * In both cases the constant field S has the algebraic property of being the residue class field K/x and K/x-c, respectively. Similar, but slightly more complicated developments hold also for the completions of K with respect to Wi(a) and W2(a), but there is no need to go into details. Consider now the valuation I alp of F and the corresponding p-adic completion Pp of F. We have m_,00
I P I p = - < !> and |c L ^ 1 for all rational integers c. It follows that every formal series Cf p* + cf+lp*"1"* + cf+2P*+2 + ••• where f and all the coefficients Cm are rational integers, converges with respect to |a|p; for the valuation Ta|p is Non-Archimedean, and It will be proved in this chapter that every element of Pp can be written in many ways as a series of this kind, but that there is one and only one series in which the coefficients assume only values in the finite set {0,1,..., p-l}. When Hensel discovered the p-adic numbers towards the end of last century, there was not yet any general field theory or theory of valuations. He defined his numbers by the series and by the rules for computing with them. In this work he followed the analogy to the Laurent series /l\f
fiY+1
f!V+2
or cf (x-c)f + cf+i(x-c)f+1 + cf+2(x-2)f+2 + ... for an analytic function in the neighbourhood of a pole, either at x=°° or at a finite point x=c. Such series are convergent in the sense of complex analysis rather than with respect to the valuations ||a|| or I la Up; but even in function theory the latter kind of convergence plays a big role in connection with the orders of poles and zeros. The investigations of this chapter are concerned only with the p-adic, g-adic, and g*-adic numbers. However, the method is much more general, and it can in particular be used to prove the earlier assertions about the completions of K with respect to ||a|| and I la Up. 1. Notation.
In this and the later chapters the notation will be essentially the same as before. Always PI,..., pr denote finitely many distinct primes, and g ^2 denotes an integer with the factorisation
28
LECTURES ON DIOPHANTINE APPROXIMATIONS g = piei... prer
where ei,..., er are positive integers. The valuations la I and I a |p, and the pseudo- valuations |a|g and |a|g*, are defined as in Chapter 1, and P, Pp, Pg, and Pg* denote the corresponding completions of the rational field F, thus are the fields of the real and the p-adic numbers, and the rings of the g-adic and the g*-adic numbers, respectively. We shall in general use Latin letters for rational numbers, small Greek letters for real and p-adic numbers, and capital Greek letters for g-adic and g*-adic numbers. If A^*-*~(cti ,..., otr) is a g-adic number with the pj-adic components a] for j=l, 2,..., r, the g-adic value of A is equal to logg logg
Similarly, if A*~~-(a, a lf ..., ar) is a g*-adic number with the real component a and the opadic components «j for j=l, 2,..., r, the g*-adic value of A* is given by
(R
logg laiig10™1 ,..., i<
logg
It will suffice to prove the first formula as the second formula may be obtained in the same way. There exists a fundamental sequence {am} in F satisfying
lim am = A (|aL) o and hence also satisfying lim am = cKi(lalpj) m—»«> J -^J These two limit formulae imply that lim l a m | g = | A | g
and
(j = 1,2,...,r). ' ' '
lim |am|pj = l«j|pj
(j = l,2,...,r).
Now, by definition,
(
log g
lamlpi
10gPl
logg ,-, la m lp?
10gPr
\ / ,
and so the assertion follows immediately. We note that (I): Un|g = (U|g)n, U*n|g* = (U*|g*)n for all AePg and A*ePg* and aU positive integers n; and that further (II): Ug n lg= for all AePg and all rational integers n. These properties follow easily from the explicit expressions for Ulg and U*|g*, and from lglg = g"1, lgl pl =Pi"ei,..., l g l p r = P r e r .
THE p-ADIC, g-ADIC, AND g*-ADIC SERIES
29
They are special cases of the properties (I) and (II) in § 4 of Chapter 1. Here and later, we have continuously to deal with limits
Urn ... (w) m—»°° where w(a) stands for one of |a|, |a|p,
|a|g, or |a|g*.
In order to shorten the formulae, we shall always omit the sign |a| of the absolute value and write
lim ... m—»°° when dealing with real limits. We further shall replace
and
lim ... (|a|p) m—x»
by
lim
by
... (|a|g)
lim ... (|aL*) m— »°°
by
lim ... (p), m— >«> lim ... (g),
m— — >o
lim ... (g*), m— ><»
respectively. This agrees with Hensel's own notation. 2. The ring I g and the ideal 9 .
Denote by Ig and g the two sets of all rational numbers satisfying lalg* 1
and
respectively; thus g is a subset of Ig, and Ig is a subset of F. The set Ig is a ring. For |a|g is a Non- Archimedean pseudo-valuation. It follows that if a and b are in Ig and hence l a | g « 1, I b l g * 1, then |a=Fb|g ^ max(|a|g, |b|g) ^ 1, l a b l g ^ |a|g|b|g
Then
alg, big) ^,
30
LECTURES ON DIOPHANTTNE APPROXIMATIONS
and hence a+b, a-b, and ac are likewise in 9 . By the identity (II), |f|g = gla|g.
Hence it follows that a belongs to 9 if and only if | belongs to Ig.
•
&
Thus 9 consists of all multiples a=a'g where a'elg. In the language of ideal theory, 9 is the principal ideal g=(g) of Ig. The elements of Ig and 9 may also be characterised as follows. Write the rational number a as a quotient a = — of two rational integers P and that are relatively prime. Then a belongs to Ig if and only if (g, Q) = 1, and a belongs to 9 if and only if g|P, (g, Q) = 1. The proof follows easily from the definition of lalg. 3. The residue class ring Ig/ 9. If a and b are two elements of Ig such that a-b lies in 9, we write
as b(9). From the ideal property of 9 one deduces easily that this is an equivalence relation: asa(9);
if a = b(0), then if a = b(0) and b = c(0), One can then subdivide Ig into classes 9+a where 0+a consists of all elements a1 of Ig such that a' = a(0). Two such classes are either disjoint, or they are identical. Denote by Ig/0 the set of all these classes. It is again easily proved that Ig/0 becomes a ring if the sum and the product of any two classes 9+a and 9+b are defined by (9+a) + (9+b) = 9+ (a+b), (9+a) • (9+b) = 9+ ab. The proof may be omitted, as this is a particular case of a well-known theorem and since similar examples have already occurred. Consider now an arbitrary element 0+c of Ig/0 . If c is written as the quotient c = 5of two integers R and S 4 0 that are relatively prime, then o
(g, S) = 1 since celg. It follows that the elementary congruence R s aS (mod g)
THE p-ADIC, g-ADIC, AND g*-ADIC SERIES
31
has integral solutions a. If a=ao is one of these, the general solution has the form
a = ao + gk where k is an arbitrary integer. There is then, in particular, a unique integral solution a satisfying 0 < a < g-1. To this solution a there exists a further integer 1 such that
R - aS = gl and hence that
R
1
Evidently
|elg
and g| e0
and therefore This relation implies further that the class 0+a is identical with the class 0+c. We have then the result that every class in Ig/0 is of the form 0+a, where a is one of the integers 0, 1,..., g-1. These g classes are all distinct. For if a and a1 are integers such that Q *£ &
A set M= {Ao, Ai,..., Ag_i} of g rational numbers is said to be a system of representatives (viz., of the classes 0+a in Ig/0 ) if Ao, Ai,..., Ag-1 lie in Ig and if Ao= 0(0), A! H 1(0),..., Ag_i= g-l(0). It follows that 0+Ao, 0+Ai,..., 0+Ag-i form again a full set of elements of Ig/0 . Such systems of representatives have the following property.
32
LECTURES ON DIOPHANTINE
APPROXIMATIONS
If A is any g-adic number satisfying and if M = {Ao, Ai ,..., Ag-a} is an arbitrary system of representatives, then there is a unique element A of M such that .
Proof: By the first chapter, F lies dense in the g-adic ring Pg and so contains a rational number c for which
and hence also |c|g = U-(A-c)|g< max(Ulg, U-c|g)
If A'eM also satisfies
then |A-Af| g = U-A') - (A-A)|g < max(U-A' |g, U-A|g) < | , so that As A1 (5), whence A=AT . 5. Series for g-adic numbers.
Denote by h4o any fixed integer such that (g, h) = 1. It is easily verified that, for all g-adic numbers A and all integers m, =g-mg
and in particular
because l h | g = |h|pl =... = |h|pr = l. Let, for every integer m, a system of representatives be given; systems belonging to different m may be equal or distinct. If A is an arbitrary g-adic number, denote by f any integer satisfying
THE p-ADIC, g-ADIC, AND g*-ADIC SERIES
33
the inequality
g- f *Ulg and put
so that By 14, there exists a unique AM
such that
Put
then
A® = A(f) + {
A^\
There again exists a unique A(f+1) of M<£+1) such that
Put
then g
^ f f U(f+l). A (f+D| &|A
A
'g
^! A
'
In this manner, we may continue indefinitely, and we so obtain for every suffix m ^f (i):
a g-adic number A^m' satisfying U( m )| g «1,
(ii):
m
m
a unique element A^ ^ of M( ) such that ^(m) = A(m) + I A(m+1>, U(m> - AM L « ± . h & g
It follows that, for all m 5* f,
= A« H-f
and
34
LECTURES ON DIOPHANTINE APPROXIMATIONS Here
~
I •» *
g
' whence
lim Therefore A' ' can be written as the convergent infinite series X® = A» + f A<*+1) + (f)2A
+ ... (g). Assume there is a second series A (f) _ a(f) . S a(f+l) , (S\ a(*+2)+ M h W for A(f) where likewise a(m)eM(m) for all m ^ f and, say, (f) (f+l) a(f)—_ A XX ', ad CL
- Att+1) ~~ XX*
',
a(m-l)
••• , «,
_ A(m-l) a(m) ± i(m) ~~ XX
, Ct*
'
T XX
'.
£
On subtracting the two series for A( ), <.«..(„„(!)»,. ^(AW-aW, (!)"(„; here, on the left-hand side, A(m)
^(m)
hence A(m)
^ a(m)(8)
and
therefore lA(m)- a (m)L> & 1 g
whence
g
w-(m+l)
On the right-hand side, A(n) and a(n) belong for all n ^ m to M^n), so that n lA^L^ 1,' |a( n )|& f f ^l,' hence |A( )-aWL
and therefore
max n=m+l This contradiction proves that the series for A® is unique. Since A = (f) A®, the following result has been obtained. Let h + 0 6e an integer prime to g; let M(m) /or ewery integer m be a system of representatives; and let A be any g-adic number. Denote by f any integer satisfying g-f ^ I A|g. 7%en A can be written in a unique way as a convergent infinite series
THE p-ADIC, g-ADIC, AND g*-ADIC SERIES A = A« (f)f
+
A»*> (f)f+1
+
35
Att*> (f) f+2 + ... (g)
where A(m)eM(m) for all m $* f. The series for A still depends on the choice of f, and entirely different series may be obtained for different values of f. Assume, however, that all systems M(m) contain the number Af,m) = 0. If A * 0, denote by f the integer satisfying so that, necessarily, f < ff. The series for A corresponding to f now becomes Mf Mf+l + ... (g) w w w vv where A< f ') =1= 0 and therefore \A fyL = g"f' |A(f) oL. In the excluded case A = Q there are only the trivial series
Finally let all systems M(m) be identical with M<m) = {0, 1, 2, ..., g-1}. The result just proved now takes the following form. Let h* 0 be an integer prime to g. Every g-adic number can be written in a unique way as a convergent series
where the coefficients A(m) are numbers 0, 1, 2,..., g-1 as follows. If A = 0, all these coefficients vanish; if A =*= 0, then it may be assumed that Ate)* 0, and then In the special case when h = 1, the corresponding series A with coefficients that assume only the values 0, 1, 2,..., g-1, is called the g-adic series for A. In particular, when g is a prime p, it follows that every p-adic number a can in a unique way be written as the p-adic series where the coefficients are integers 0, 1, 2,..., p-1; for of = 0 all these coefficients are zero, while for a £ 0 it may be assumed that A(f) * 0 and then la In = P"f
because
|A(*)|D = 1.
36
LECTURES ON DIOPHANTINE APPROXIMATIONS
All the developments for the g-adic number A studied in this section have one important property in common. If is the decomposition of A into its pi-adic,..., pr-adic components, then the same series converge also simultaneously to the limits «i,..., ar with respect to the valuations la|pl,..., |a|pr, respectively. Of particular importance are those g-adic numbers A which satisfy the inequality and are called g-adic integers. Their g-adic series do not contain non-zero terms A(m)gm with negative exponents m. A proof just like that for the set Ig in § 2 shows that the set Jg, say, of all g-adic integers forms a ring. In the same way, the p-adic integers a are defined by the inequality Their ring Jp is a domain of integrity, and it has Pp as its quotient field. The g-adic series for A and its special case, the p-adic series for a, are extremely useful for actual computations with such numbers. The technique of such computations is explained, with many examples, in HensePs book "Zahlentheorie". We conclude this section with a nearly trivial remark. There are infinitely many distinct g-adic numbers i,..., ar) with one and the same first component on , but with varying other components, at least if r ^ 2. Each of these numbers can be written as a g-adic series A = A<«gf + A(f+Dgf+l + A(f+2)gf+2 + ... (g), and then the same series converges to its first component ai with respect to the valuation la!Pl . Furthermore, different g-adic numbers naturally are represented by different g-adic series. Hence there are infinitely many distinct g-adic series + A(f+2)gf+2 + . for any given pi-adic number, provided only that g is divisible by at least one further prime distinct from pi . * 6. Series for g*-adic numbers.
The results obtained in§ 5 lead also to convergent series for g*-adic numbers It is convenient to decompose such numbers into their real and g-adic components and to write A*— (a, A)
THE p-ADIC, g-ADIC, AND g*-ADIC SERIES
37
where A++(ai, ..., ar). As before, the integer h * 0 is assumed to be relatively prime to g; in addition, let
Kl < '• Further, if p is any real number, denote by M(p) that system of representatives which consists of the integers k satisfying p - g
and put ^*(f)
=
(I) 4*^a(f),A(*)), where A(f)-^(a(f),..., a$>). \h/
Then
and hence By §4, there exists a unique A(f)eM(o?(f)) satisfying UW-AW|g* J; from this definition, also att) - g = ({j)" ^*® - A(«)-(«M),A(fH-l)), where AM)^(« so that A® + f A*(f^),
o»+« = (f y^a ® - A(fy ^(f+D = (f
and therefore
By the same reasoning as a few lines back there further exists a unique AW+l)eM(att+D) satisfying A(f+D |g * i . D
38
LECTURES ON DIOPHANHNE APPROXIMATIONS
On putting A*(f+2) = ^S]" 1(^*(f+l) _ A(f+1))^(a(f+2), Aft+2)), ^(f+2)^^(ajf+2),..., <xrf+2)), it follows that
whence This construction can be repeated indefinitely, and we so obtain for every suffix m ^ f (i): a g*-adic number A*(m)^(a(m)yA(m)) satisfying (ii):
Ia ( m ) l
and
0 ^ a < m ) - A(m)
Then for all m ^ f, A*(f) = Att) + ^ Aft+l) + f- )2 A^+2) + ... + ff J " " Here - f a< m )|
ir^ m - f ^)|g^g-(m-f),
so that both expressions tend to 0 as m-^«>. It follows that
f
"•»i(sr
Hence A*( ) can be written as the convergent series = Ad) + Att+« + 2 and since A* = | T A*^), it also follows that
A(f+2)+
... (g.)f
THE p-ADIC, g-ADIC, AND g*-ADIC SERIES
39
In these two series the coefficients A(m) are integers such that 0 < a(m) . AM < g, where \a^\< |h| . Their absolute values are therefore smaller than g+|h| , and hence |A(m)|< g + |h|- 1 for m ^ f . In the special case when h is positive and when a & 0, it is easily deduced from the definition of the real numbers a(m) that all these numbers are nonnegative. We therefore find now that -g ^ a(m) „ g < A(m) «* oM < h and hence -(g-1) < AM < h-1. Thus the following result has been obtained. Let h * 0 be an integer prime to g such that |j \< 1, and let A*-*+-(ci,A} be any g*-adic number. Denote by f any integer satisfying
Then A* can be written as a convergent infinite series A* = A(f)(f)£ + A(M)(f)f+1 + A«+2)(f)£+2 + ... (g*) where all coefficients A are integers at most of absolute values g+|h|-l. If h >0 and as*0, it may be assumed that these coefficients satisfy the stronger condition - (g-1) < A(m) ^ h-1. A look through the proof shows that this result remains valid even when A* reduces to its real component a, i.e., when g = 1, r = 0, and there is no g-adic component. We then obtain the classical representation of the real number a to the basis h, with "digits" A(m) that assume only the integral values 0, 1, 2, ..., h-1. In the g-adic case the set Jg of all g-adic integers, i.e., of all g-adic numbers A satisfying Ulg ^ 1, was found to be a ring, due to |a|g being Non- Archimedean. In addition, the g-adic aeries for the elements of Jg consisted of terms A(m)gm that were rational integers; hence g-adic integers have the characteristic property of being limits of sequences of rational integers. It might seem appropriate to introduce also the set Jg* of all g*-adic numbers -4* satisfying |A*fg* « 1 and to call its elements g*-adic integers. However, since |a|g* is now Archimedean, Jg* is no longer a ring, but is closed only under multiplication. Moreover, the elements of Jg* distinct from 0, +1, and -1 cannot be approximated arbitrarily closely by sequences of rational integers. There is thus no justification for singling out the elements of Jg* among other g*-adic numbers. The series studied in this chapter give explicit expressions for the elements of P, Pp, Pg, and Pg*, and their finite sections give rational
40
LECTURES ON DIOPHANTINE APPROXIMATIONS
approximations. The same series will be applied in Chapter 4 to the construction of continued fractions for g-adic and g*-adic numbers. 7. Sequences that converge with respect to all valuations of F.
Let {pm} be the sequence of all prime numbers 2, 3, 5,... written in ascending order, and let a be a real number, while <XM> for M = l , 2, 3,..., is a pM-adic number. We conclude the chapter with the following simple remark. There exists an infinite sequence {am} of rational numbers such that, simultaneously, lim a = «; lim am = aM(PM) for all M. m—»°o m m—>°° For, as we saw, the different valuations are independent and hence it is possible to find, for each suffix m, a rational number am such that lam - « ! < — ; lam - «i lp!<~ , lam - «2 Ip2<— ,—, lam-^mlpm^' Then the sequence {am} has the required properties. A similar proof shows that if each <XM is a pM-adic integer, a sequence {am} of rational integers can be found such that lim am = <XM(PM) for a11 MThis result is easily seen to be equivalent to the following one. There exists an infinite sequence {Am} of rational integers satisfying 0 < Am < m such that, simultaneously* Ai'l! + A a - 2 ! + A 3 -3I + ... = «M
Chapter 3 A TEST FOR ALGEBRAIC OR TRANSCENDENTAL NUMBERS Suppose a field K has been extended to its completion KW with respect to some valuation or pseudo- valuation w(a). The element a of Kw is then said to be algebraic over K if it satisfies an equation aoxn H- aix11"1 + ... + an = 0 (ao * 0, n 5* 1) with coefficients in K, and it is otherwise called transcendental over K. The corresponding extension field K(a) is likewise algebraic, or transcendental, respectively. Much of the following investigations are concerned with the problem whether a given real, p-adic, g-adic, or g*-adic number a is algebraic or transcendental over the rational field r . There is one, relatively elementary, approach to this problem where one studies the values of variable polynomials F(x) with rational integral coefficients at the point x = a; it will be studied in the present chapter. A much deeper method, due to Thue, Siegel, and Roth, uses properties of the rational approximations of a. For the explicit construction of such approximations a simple algorithm will be given in the next chapter. It is based on the continued fraction algorithm for real numbers and forms its natural extension to p-adic, g-adic, and g*-adic numbers. The deep theorem of Roth shows that these approximations cannot be too good in the case of an algebraic number a. The theorem is bestpossible and has many interesting consequences. The long and involved proof fills all the remaining chapters. Before starting with these investigations, it has perhaps some interest to collect certain general properties that are basic for the theory of Diophantine approximations. One such property was already mentioned in the first chapter. This was the Fundamental Inequality: la I n |a|p. where a 4s 0 is any rational integer, and pi,..., Pr are finitely many distinct primes. Thus, in particular, |a|>l
and
|a|p * -j^j- .
The second property makes a statement on the density of the rational integers on the real axis: If a. and |3 are real numbers such that a < /3, then there are exactly [p] - [a] rational integers g such that a < g ^j3. In particular, the interval 0 ^ g ^ ]3 contains exactly [|3] + 1, and the interval -)3 ^ g ^ /3 exactly 2[/3] + 1 rational integers. 41
42
LECTURES ON DIOPHANTINE APPROXIMATIONS
Here [a] denotes as usual the integral part of a, i.e., the greatest rational integer that does not exceed a. The third property is the famous Principle ofDirichlet (Schubfachprinzip): If n+1 elements are distributed among n sets, then at least one set contains more than one element. As we shall find, all three properties will again and again be applied on the following pages. 1. Notation. Let aeJP, a0ePp, -AePg, and A*ePg* be any real, p-adic, g-adic, and g*-adic number, respectively, and let la|, I alp, |a|g, and |a|g* be the corresponding valuations or pseudo-valuations. When properties in common to all four kinds of numbers are being discussed, we shall use the letter a for any one of them; and the symbol co(a) will then stand for the corresponding valuation or pseudo-valuation. By integer, without any qualifying term, we always mean a rational integer. All polynomials that occur in this chapter lie in the polynomial ring r[x], thus have rational coefficients. If g(x) = boxq + bixq~ + ... + bq is such a polynomial, the maximum IgSJl =max(|bol, Ibil,..., |bq|) is called the height of g(x). Throughout this chapter, F(x) = Aoxm + Aixm-1 + ... + Am is an arbitrary polynomial with integral coefficients, of a degree not exceeding m and of height A = |F(x)|. 2. The minimum polynomial of an algebraic number. Let a be a fixed number (real, p-adic, g-adic, or g*-adic) which is algebraic over F. There exists thus at least one polynomial g(x) * 0 in r [x] such that g(a) = 0. Among all polynomials with this property we select one of lowest degree, the polynomial f(x) = aoxn + aix11"1 + ... + an, where ao * 0, n ^ 1, say.
If g(x) vanishes for x=a, then g(x) is divisible by f(x). For g(x) may be divided by f (x) and then takes the form g(x) = f(x)h(x)+k(x), where h(x) and k(x) are again in r[x], and k(x) is of lower degree than
A TEST FOR ALGEBRAIC OR TRANSCENDENTAL NUMBERS
43
f(x) or vanishes identically. Since k(a) = g(a) - f ( a ) h (a) = 0, k(x) is divisible by f(x), whence k(x) = 0. We call f (x) the minimum polynomial, and f (x) = 0 the minimum equation, for a. Except for a constant factor distinct from zero, the minimum polynomial evidently is unique, and hence so is its degree n. The number a is then likewise said to have the degree n. If the algebraic number a is real or p-adic, then the minimum polynomial f (x) is irreducible over F, but this need not be true if a is a g-adic or g*-adic\number. First let a belong to either P or Pp; thus a is element of a field. Further suppose that f (x) = g(x)h(x) where g(x) and h(x) are nonconstant polynomials in r[x] and therefore are of lower degrees than f (x). It follows that g(a) +0 and h(a) + 0, hence that also the product gta)h(a) = f ( a ) is distinct from zero, which is false. Secondly assume that a lies in Pg or Pg*. Then the last proof is no longer valid since there are zero divisors in both rings. By way of example, the g-adic number -A-**-(l, 0,..., 0) has the minimum polynomial f(x) = x(x-l) which is reducible over F[x], A g-adic or g*-adic number is algebraic if and only if all its components are algebraic. The proof is the same in both cases; it suffices therefore to deal with the case of a g-adic number A-^tei,..., «r). K A is algebraic, let f(x) be its minimum polynomial. Then f(A)—[ftei),..., f(a r )] = 0, hence f(«i) = ... = f(a r ) = 0. It follows that all components ai,..., ar are algebraic, and that their minimum polynomials are divisors of f(x). Conversely, let ai,..., ar be algebraic, with the minimum polynomials fi (x),..., fr(x), respectively, and let f (x) be the least common multiple of these polynomials. Then f(o?i) = ... = f(a r ) = 0 and hence fU)—[f(ai),..., f(a r )] = 0 and so A is likewise algebraic. It is clear from this proof that the minimum polynomial of A is equal to the least common multiple of the minimum polynomials of its components. The same is true for algebraic g*-adic numbers A*. The minimum polynomial of an algebraic number or has no multiple zeros, hence is relatively prime to its derivative. The assertion certainly holds when a = a or a = a0 because then f (x) is irreducible over F, a field of characteristic 0. It still remains valid when a = A or a = A* because the least common multiple of finitely many polynomials irreducible over F cannot have multiple zeros. 3. An algebraic identity.
Let a be again an algebraic number. We may assume that its minimum polynomial
44
LECTURES ON DIOPHANTINE APPROXIMATIONS f(x) = aoxn + aix11"1 + ... + an, where ao * 0, n ^ 1,
has integral coefficients, and we write Denote by F(x) = Aoxm + Aixm-! + ... + Am a second polynomial with integral coefficients, of height We assume, for the present, that (i): Ao * 0, so that F(x) has the exact degree m; and (ii): F(x) is relatively prime to f(x). These restrictions will later be relaxed. As is proved in algebra, these two conditions imply that the resultant
R=
a0
ai
a2 ...
a n _i
an
0
a0
ax
a n _2
*n-l
0 A!
0 ... a0 ai A2 ... Am_! Am 0
AQ
A1 ...
...
0
...
0
an ...
0
a2 ... ...
an 0
Am-2 Ajn.i Am
...
m rows
0 n rows
0
0
...
AQ
A!
A2
...
Am
of f (x) and F(x) is distinct from zero. Denote by g(x) and G(x) the two determinants that are obtained from the determinant for R by leaving the first m+n-1 columns unchanged, while replacing the last column by a new one consisting of the successive terms m-2 ,..., 100 0 xm-l , xA. A A, U, U, ..., \J and
0, 0, ..., 0, xn-1, xn-2, ..., 1,
respectively. Then the following identity holds: (1): R = f (x)g(x) + F(x)G(x). For multiply the 1st, 2nd,..., (m+n-l)st column of R by the factors x m+n-l j and add to the last column. This operation does not change the value of the determinant, but transforms it into a new determinant where the last column now consists of the terms X m-lf( x ),
x»-2fW,...f l-f(x), x*-lF(x), xn-2F(x),..., On splitting this column into the two columns
A TEST FOR ALGEBRAIC OR TRANSCENDENTAL NUMBERS X m-lf(x),
45
xm-2f«, ..., l.f( x ), 0, 0,... 0,
and
0, 0, ... 0, x*-lF(x), xn-2F(x), ... , l-F(x), the assertion follows at once. From its definition as a determinant of order m+n, R is a sum of at most (m+n)! terms where each term is a positive or negative product of m factors a^ and n factors A^ . It follows that R satisfies the inequality (2): 1* |R|« (m+n)! Here the lower bound holds because R is an integer distinct from zero. Similarly, g(x) and G(x) may be written as polynomials of the form g(x) = box™-1 + bixm-2 + ... +b m -l and G(x) = Box^-l + BiXn-2 + ... + Bn-1, where the coefficients bju, and Bv are the cofactors of the last column of R, hence are determinants of order m+n-1. They are thus integers, and an estimate just as for R leads to the inequalities HiWl * (m+n- 1) 1 am-
(3): (4):
4. Inequalities for algebraic numbers.
On putting x = a in the identity (1), it follows that R =F(a)G(a) and hence that (5):
In order to deduce lower bounds for co{F(a)} from this equation, it becomes necessary to distinguish between the four different kinds of numbers. Case I: a = ot is a real algebraic number. Since G(x) is at most of degree n-1, the inequality (4) implies that |a| n - 1 +|a| n - 2 +...+ W+l) ^ lal11-1 + |a|n"2 + ...+ |a| + l). On the other hand, by (2), |R|*1. It follows therefore from (5) that (1,1): where the factor Cl (m) = depends on a and m, but is independent of A.
46
LECTURES ON DIOPHANTINE APPROXIMATIONS
Case 2: a = a0 is a p-adic algebraic number. Since the polynomial G(x) has integral coefficients and is at most of degree n-1, iGteojIp^maxOaol*.- 1 , l«olg'2,..., |a0|p, 1) = maxdaol*"1, 1). On the other hand, by (2) and the fundamental inequality, Hence it follows from (5) that (1,2): |F(a 0 )l p 5*c 2 (m)A- n where the factor c a (m) = {(m+n)I a^maxdaolg- 1 , I)}"1 depends on aQ and m, but is independent of A. So far, the inequalities (1,1) and (1,2) have only been proved if F(x) satisfies the conditions (i) and (ii) of § 3, • (i): F(x) has the exact degree m, and (ii): F(x) is relatively prime to f (x). However, both inequalities remain valid if only the following weaker conditions are imposed, (if): F(x) is at most of degree m, and (ii1): F(a) * 0. For let mf be the exact degree of F(x). Then m' < m, hence ci (mT ) ^ ci (m), c2 (m1 ) ^ c2 (m). The inequalities (1,1) and (1,2) corresponding to the degree mf are thus at least as strong as those corresponding to the degree m and imply the latter. Next, in both cases a = a and a = «o the hypothesis F(a) * 0 implies that F(x) is relatively prime to f(x). For the polynomial f(x) is now irreducible over T, and so F(x) and f (x) cannot have a nonconstant common factor unless F(x) is divisible by f(x). Case 3: a = A-~-(ai,..., ar) is a g-adic algebraic number. Let again F(x) be at most of degree m, and let F(A) * 0. Then )— [F(ai),..., F(ar)] * 0, and so there is a suffix j with 1 < j ^ r such that F (aj) * 0. Let f j(x), the minimum polynomial of aj, have the degree n(j) and the height a(j). The inequality (1,2) maybe applied to the component F(OJ) of F(A), giving where c2(i)(m) stands for |aj|^i)'1, I)}"1. Put
A TEST FOR ALGEBRAIC OR TRANSCENDENTAL NUMBERS
47
and
/ logg logg \ eil gPl Cs(m) = min UW) ° , ...,caW(m)erlogPr/. By the definition of the g-adic value, logg |F(A)|g = max It follows then, finally, that (1,3): where the exponent n(A) is a positive number depending only on A, and c8(m) is a positive number that depends on A and m, but not on A. Case 4: a = A* is a g*-adic algebraic number. Let A**->~(a, A) be the decomposition of A* into its real and its g-adic components, We assume that F(A*) t 0. Since at least one of the two components F(o?) and F(A) is distinct from zero. If F(a) + 0, then |F(a) | satisfies an inequality of the form (1,1); if, however, F(A) * 0, then an inequality of the form (1,3) holds for |F(A) |g. Now, by the definition of the g*-adic value, |F(A*)| g *=max(|F(a), |F(A)|g). It follows therefore that there exist a non-negative number n(A*) depending only on A* , and a positive number c4(m) depending on both A* and m, but not on A, such that (1,4): |F(A*) Ig* > c4(m)A-n(^*)On combining the four inequalities (1,1)- (1,4), the following result is obtained. Theorem 1: Let a be an algebraic number (real, p-adic, g-adic, or g*-adic), and let F(x) by a polynomial with integral coefficients, of degree at most m and of height A. There exist a non-negative number n(a) depending on a but not on m or A, and a positive number c(a, m) depending on a and m but not on A, such that either F(a) = 0, or w{F(a)} >c(a,m)A- n ( a ). The importance of this theorem lies in the fact that the exponent n(a) is entirely independent of the polynomial F(x). As we shall see, the position is very different for transcendental numbers. There is a second theorem involving the product of the values of the components of F(a). Put
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LECTURES ON DIOPHANTINE APPROXIMATIONS
|F(a0)lDp
if a= «o,
rr ]
if a = A*M-(<XI,..., a r ),
i=
,_ ^_,, ,„,
if a = A*-**"C
A proof just like that of Theorem 1 leads to the following result. Theorem I1: Let the hypothesis be as in Theorem 1. Put v(a) equal to n-1 or n, according as a has, or has not, a real component; here n denotes the degree of a. There exists a positive number r (a, m) depending on a and m but not on A such that either ft{F(a)}= 0, or fl{F(ct)} > y(a, mjA-^a). The proofs of both theorems depend very essentially on the Fundamental Inequality. The other two general properties mentioned in the introduction have not been used. They form the basis for the next investigations. 5. A theorem on linear forms.
From now on the number a need no longer be algebraic. Our aim is to find polynomials F(x) for which o>{F(a)} is small. The construction makes use of Dirichlet's principle and of the density properties of the integers which were mentioned in the introduction. In the special case when F(x) is a linear polynomial, a simpler and more explicit method will be given in the * next chapter. We begin'with a general theorem on linear forms1 Theorem 2: Let
n Lh(x) = u ahkxk (h = 1,2,..., n) k=l be n linear forms in n variables with real coefficients, and let n a= max £ lahkl>0h=l,2,...,n k=i
If Xi,..., Xn are n positive numbers such that Xi...X n > a11, there exist n integers xi,..., xn not all zero satisfying |L h (x)|<X n
(h = 1,2,..., n).
1. This is a slightly weakened form of Minkowski's theorem on linear forms (Geometrie der Zahlen, §§ 36-37). I learned the proof as given here more than 30 years ago from my teacher C. L. Siegel.
A TEST FOR ALGEBRAIC OR TRANSCENDENTAL NUMBERS
49
Proof: Denote by N a very large positive integer, and define n further positive integers ti,..., tn by the formulae
t h -LTAhT J
+1
(h = 1,2,..., n).
Since, asymptotically, ti... tn - . a" (2N+l)n AI .. .AH
as N -» oo,
N can be chosen so large that ti...t n < (2N+l)n. The ordered system (xi,..., xn) is said to be admissible if each xh equals one of the 2N+1 integers 0,71,72,..., 7N between -N and +N. There are then (2N+l)n admissible systems. For such systems, n n |Ln(x)| = I E a h k x k l ^ N E l a f c k l ^ a N , k=l k=l
so that -aN ^ Lh(x) ^ +aN.
Divide the interval [-aN, +aN], for each suffix h=l, 2,..., n, into th subintervals of equal length y^, the subintervals j[h\ J^\...9 j£', say; points on the boundary of two adjacent intervals should be counted as belonging to only one of them. In accordance with the values assumed by the forms Li(x),..., Ln(x), there corresponds to each admissible system (xi,..., xn) (1) (n) (h) a unique ordered system (Jj ,..., Jj[n ) of n subintervals such that Lh(x)eJ£ . By the construction, the number of all such systems of n subintervals is exactly ti,... tn, hence is less than the number of admissible systems (xi,...,xn). It follows then, from Dirichlet's principle, that there exist two distinct admissible systems (xi,..., xn) and (xi',..., xtf) for which the values of Li (x'),..., Ln(x') and of Li(x"),..., Ln(x") lie in the same system of subintervals (Ji7,..., J{ L A 0- Therefore l n (h = 1,2,..., n). *•»!
Put f
tf
T
.
It
Xi = Xi - Xi ,..., Xn = X!n - Xn.
Then xi,..., xn are integers not all zero for which |Lh(x)|= |Lh(x!) - Lh(x")|«s -T—
(h=l,2,... n)
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LECTURES ON DIOPHANTJNE APPROXIMATIONS
because 1> —
6. On a system of both real and p-adic linear forms. From Theorem 2 we shall now deduce a result on the values of systems of linear forms that have coefficients in different completions of the rational field. Denote by a linear form with real coefficients not all zero for which further by pi ,..., pr finitely many distinct primes, and by lj(x) = «j1x1+...+«jMxM a linear form with pj-adic coefficients satisfying
(] = 1,2,..., r)
maxdaj! lpj,..., I«JM!PJ) * 1 (J = M,.,., *)• Since l(x) has at least one non-zero coefficient, there is no loss of generality in assuming that aM* 0;
for, if necessary, it suffices to renumber the variables. Let Ei,..., Er be r arbitrary positive integers. By hypothesis, the coefficients aj^ of each form lj(x) are p^-adic integers. Hence there exist rational integers aj ^ satisfying the inequalities
Now put
ajLx1+ ... + ajM XM +
Lj(x)
pf 1 xM+j}
(j= 1,2,..., r)
if j = 1,2,..., r, if j = r+pi, /!= 1,2,..., M-l, if j = M+r
X][jl
l(x)
The linear forms Lj(x) so defined have real coefficients such that the sum of the absolute values of the coefficients of each form is not greater than 1. Therefore, on applying Theorem 2, with n=M+r and a=l, it follows that there exist integers xi , x2 ,..., XM+r n°t oil zero satisfying the system of inequalities |Lj(x)| < Aj
(j = l,2,...,M+r),
A TEST FOR ALGEBRAIC OR TRANSCENDENTAL NUMBERS whenever Xi , Xa ,..., A-M+r
are
51
positive numbers such that
Let us now specialize this result in two different ways, First specialization: Denote by T a number greater than 1 and define a number t > 1 by tM-l further put if j = 1,2,..., r, i f j -p+ttji.l,2,...,M-l,
*J 4
HJ
Then and so we may apply the last result. It follows then from the formulae for Lj(x) and Xj that there exist integers xi,x2,..., XM+r not all zero satisfying the inequalities llfto+pfJxM+jk I (j = 1,2,..., r), Ixj< t
( M = 1,2,..., M-l),
The first r of these formulae imply that already xi, ..., XM cannot all vanish; for otherwise also XM+I = ... = XM+I. = 0. These first r inequalities are equivalent to
because the expressions on the left-hand sides are integers. Hence > = 0(modpj E J)
(j = 1,2,..., r)
and therefore ll](x)lpj« P ] - E J
(j = 1,2,..., r).
Now M
= I E so that also
(j = 1,2,..., r).
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LECTURES ON DIOPHANTINE APPROXIMATIONS
Finally, from T >1 and t > 1 and from the last M inequalities, we deduce that
Since |<XM! * 1, it follows that nx
/I I I \\ ^ •• max(|xi I,..., IxMlX i *» aMi •
Second specialisation: Put Kx)=x M which is evidently allowed, and choose T=J
where tM = 2(M+l)r p?1... pEr .
Otherwise leave the notation just as in the first specialisation. On repeating the last computations, it now follows that there exist integers xi,..., XM not all zero such that |lj(x)|pj< P : E J a = 1,2,..., r), maxdxil,..., Ixjvjl) < t. The two results just proved contain the following theorem. Theorem 3: Let l(x) = «ixi+...+ «MxM, where 0< \a\ I+...+|«M| < 1, be a linear form with real coefficients; let PI ,..., pr be finitely many distinct primes; let lj(x) = ofjlXl+...+ ajMxM, where max(|aj1lp],...,|ajM|pj) < 1, be, for j = 1,2,..., r, a linear form with pj-adic coefficients; let E!,...,Er be positive integers; and let T> 1. (i): There exists a positive constant Ai independent of Ei,..., Er, and T, such that there are integers xi,..., XM for which -Ei it / \i — ~E-|. px ^ PI ,•••> H r W l p r ^ P r »
0<
(ii): There exists a positive constant A2 independent of Ex ,..., Er such that there are integers xi ,..., XM for which pl
*PrEl,..., llr(x)lp r ^p r E r , A2(p!El...
A TEST FOR ALGEBRAIC OR TRANSCENDENTAL NUMBERS
53
Remark: Theorem 3 may be extended to systems that contain more than one real linear form, more than one pi-adic linear form, etc., and more than one pr-adic linear form. The best method is to apply either Minkowski's theorem on linear forms or his theorem on lattice points in convex bodies. 7. Polynomials F(x) for which w {F(a)} is small.
Let a be an arbitrary algebraic or transcendental number (real, p-adic, g-adic, or g*-adic). Theorem 3 leads to the construction of polynomials F(x) = Aoxm + Aixm-1 + ... + Am with integral coefficients, of degree at most m and of height A > 0, for which co{F(a)} is small. They are obtained by specialising the linear forms l(x) and lj(x) and choosing the parameters Ei,..., Er, and T suitably. The theorem is applied with M=m+l, and with Ao, Ai,..., Am instead of XI,...,XM as the variables. One must again distinguish between the different kinds of numbers. Case 1: a = a is a real number. In Theorem 3(i) choose r=0 and l(x) = K~l(h><*m + Aid!111-1 + ... + Am) where /c= |a| m + \a\m-l + ... + H + l. Further denote by s an arbitrary number greater than 1 and put
T = KSm. Since the hypothesis of the theorem evidently is satisfied, it follows that there exists a polynomial F(x) for which (IV,1):
|F(ff ) | ^ s-m,
0 < A < A!/c1/m-s.
Cases 2 and 3: a is a p-adic or a g-adic number. Let a = A-*-^(a?i,..., ar) be a g-adic number; for r=l this includes the case of a p-adic number. Define non- negative integers fi ,..., fr by the equations pjj = max(l, |0j| pj )
(j = 1,2,..., r)
and apply Theorem 3(ii) to the linear forms "1 + ... + Am)
(j = l,2,...,r)
the coefficients of which obviously are pj-adic integers. Also put Ej = mfj + (m+l)ejt
(j = 1,2,..., r)
where t is an arbitrarily large positive integer and ei ,..., er are, as usual, the exponents in ei er g = Pi -Pr -
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LECTURES ON DIOPHANTINE APPROXIMATIONS
The hypothesis of the theorem is again satisfied. It follows that there exists a polynomial F(x) with the properties (IV,2):
/ r \ m ^, Q< &*$ te ( A Pjj) m+1 • g* .
m+1
|F(A) Ig « g""^
For we first obtain the second inequality together with the formulae IrCapIp,*^*1^* (j = 1,2,..., r), and these imply the first inequality by the definition of the g-adic value. Let, in particular, a = a0 be a p-adic number. We now define a single non-negative integer f by and apply the inequality (V,2) with g=p and r=l. It follows that if t is an arbitrarily large positive integer, there exists a polynomial F(x) with the properties fm (m+1)t (IV,3): |F(nd)l p * p" , 0< A * A2pm+1 -pt. Case 4: ,a = A*—-(a, oti,..., ar) is a g*-adic number. Define K, fi,..., fr, l(x), li (x),..., lr(x) exactly as in the first two cases, and put T = ftgmt; EJ = m(fj+ejt) (j = 1,2,..., r), where t is again an arbitrarily large positive integer. The hypothesis of Theorem 3(i) is then satisfied, and it follows that there exists a polynomial F(x) for which 1 j* (IV,4): |F(A*) |g* « ff mt , 0 < A « A! «m- ^ pj • g2t. FOB the second inequality is obtained exactly in the form given, but instead of the first inequality we obtain the r+1 relations I F(«) I * g-mt, I F(aj) U « Pj-^mt These, however, imply that
(j = 1,2,..., r).
|F(A) Ig ^ g'mt, hence that max(|F(a) |, |FU) |g) < g-mt f so that the assertion follows from the definition of the g*-adic value. In the four cases denote by u the upper bounds \ m fm L
respectively, that were obtained for A. The results just proved may be combined to give the following theorem.
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55
Theorem 4: Let a be an arbitrary number {real, p-adic, g-adic, or g*-adic), and let m be any positive integer. Put m if a is real, m+1 if a is p-adic or g-adic, y
if a is g*-adic,
There exists a positive constant y(a, m) depending on a and on m but not on A, as follows. If u is any sufficiently large positive number, there is a polynomial F(x) with integral coefficients, of degree at most m and of height A, such that co{F(a)} «sy (a,m) u"M(a»m), 0 < A ^ u. Hence also This theorem has again an analogue for the function £2{F(a)}, defined in § 4, which is proved just like Theorem 4. Theorem 41 : Let a and m be as in Theorem 4, and let ( m if a is real or g*-adic, M(a,m) = if Q .g p_ There exists a positive constant r(a,m) depending on a and on m but not on A, as follows. If u is any sufficiently large positive number, there is a polynomial F(x) with integral coefficients, of degree at most m and of height A, such that n{F(a)} ^ r(a,m)u"M(a'm), 0 < A ^ u. Therefore also
«{F(a)}^r(a,m)A" M(a ' m) . 8. A necessary and sufficient condition for transcendency.
From now on let a be a transcendental number. The polynomial f (x) of Theorem 4 then satisfies the inequalities 0 < o>{F(a)} « y (a,m)u" fl( * m) , 0 < A ^ u, because F(a) * 0 by the hypothesis. Assume further that the parameter u tends to infinity, so that co{F(a) } tends to zero. This implies that F(x) cannot remain fixed, but must run over an infinite sequence of distinct polynomials with heights A tending to infinity. For there are only finitely many polynomials F(x) with integral. coefficients, of degrees not greater than m and of bounded heights; and for such a set of polynomials co{F(a)} necessarily has a positive minimum. Hence, if a is transcendental, there exist infinitely many distinct
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LECTURES ON DIOPHANTINE APPROXIMATIONS
polynomials F(x) with integral coefficients, of degrees not exceeding m and of heights tending to infinity, such that 0 < o>{F(a)} This is true whatever the value of m. We may then choose m so large that jjt(d,m) is greater than any prescribed positive number A and that therefore as soon as A is sufficiently large. This result may be combined with Theorem 1, leading to the following test for transcendency. Theorem 5: The number a (which may be real, p-adic, g-adic, or g*-adic) is transcendental if and only if, given any positive number A, there exist a positive integer m and infinitely many distinct polynomials F(x) with integral coefficients, of degrees not exceeding m and of heights A tending to infinity, such that 0 < o>{F(a)} ^ A-A. This test leads immediately to a special class of transcendental numbers, the Liouville numbers2 . A number a is said to be a Liouville number if there exist, (i) an infinite sequence of distinct rational numbers l, Hn = max(|Pn|,|Qn|), and (ii) an infinite sequence of positive numbers {Ai, A2, A3,...} tending to infinity, such that 0
(n = 1,2,3,...).
Every Liouville number is transcendental. For put Fn(x) = Qux-Pn so that FQ(X) is a polynomial of the first degree of height Hn. Then
because, by the definition of The assertion is thus contained in the special case m=l of Theorem 5. There is no difficulty in constructing Liouville numbers, and thus transcendental numbers, of each of the four kinds. Thus the real number 2. The real Liouville numbers were discovered by Liouville in 1844; for the reference see the introduction to the second part. These numbers gave the first explicit examples of transcendental numbers.
A TEST FOR ALGEBRAIC OR TRANSCENDENTAL NUMBERS 00
00
00
57 .
2v
.
£ 2- nI , the g-adic number £ S nl , and the g*-adic number £ g+ rTf) 1 1 1 \ V all are Liouville numbers. This is proved easily by taking as rational approximations 7^- the sums of the first n terms of these series. Wn However, Liouville numbers are the least interesting examples of transcendental numbers. None of the more important constants of real analysis, like e,?r, and log 2, are Liouville numbers, although they are transcendental. Actually, proofs of their transcendency may be based of Theorem 5, but it then is necessary to consider polynomials F(x) of arbitrarily high degree. We shall later prove some theorems by means of which it is possible to construct transcendental numbers that are not Liouville numbers. But our aim is not to give a general theory of transcendental numbers and of proofs of transcendency. For this important and beautiful theory the reader is referred to the following three books: A. Gelfond, Transcendental and algebraic numbers (in Russian), Moscow 1952. C. L. Siegel, Transcendental numbers, Princeton 1949. Th. Schneider, Transzendente Zahlen, Berlin 1957.
Chapter 4 CONTINUED FRACTIONS Theorem 4 of the last chapter shows, in the special case when m=l, that there exist pairs of rational integers P, O not both zero for which co(Qa- P) is arbitrarily small. The proof of the theorem, and that of Theorem 3 on which Theorem 4 is based, actually allows the effective construction of such approximations. However, the many pairs P, O that have to be considered in order to find one satisfactory pair make this method prohibitive on account of the amount of labour that is required. Free from this defect is a method based on the algorithm of continued fractions which is to be discussed in the present chapter. The theory of continued fractions may go back to the times of classical Greek mathematics, as far as the real case is concerned; that for p-adic, g-adic, and g*-adic numbers, on the other hand, seems to be new in the form given here. Implicitly, they occur, however, already in an old paper of mine1. We shall begin with a short treatment of the regular continued fractions for real numbers. For further details the reader is referred to the standard works on the subject, e.g. to that by O. Perron. The continued fractions for p-adic, g-adic, and g*-adic numbers are then derived by means of a simple idea that makes use of the series for such numbers studied in Chapter 2. 1. The continued fraction algorithm in the real case.
Let da be any real number. Put ao = [«0], so that ao < ao < ao + 1. If a0 is an integer and hence a0 = ao, this ends the algorithm. Otherwise put ao - ao + —, where evidently ai > 1. Put again ai = [0i ], so that ai ^ «i< ai + 1. If now ai is an integer, then oti = ai, and the algorithm again breaks off. In this manner we can continue. Either the algorith finally ends when we reach a number an which is an integer; or this never happens, and then the algorithm may be continued indefinitely. After n steps, it consists of the formulae, a0 = ao + — , where ao = |>o], «i > 1, 1
Znr Approximation algebraischer Zahlen III, (1934), Acta math. 62, 91-166. See, in particular, the first part of this paper. 58
CONTINUED FRACTIONS
59
on = ai + —, where ai = [cti ], aa > 1, «2
•
an-l = a n _i + —, where a n _i = [an-lL «n > 1. For shortness, we express this set of n formulae in the abbreviated form «o = [ao,ai,..., an-l, «n] which stands for the explicit formula 1 = ao +
a2
obtained by eliminating HI, Q!2,..., an-lIf the algorithm terminates with the integer an, then Qfn = an and ao = [ao, ai ,..., an»i, an]. We call the symbol on the right-hand side a, finite continued fraction for a0. If, however, the algorithm never breaks off, then we write ao = [ao,ai, fe ,...], and say that the symbol on the right-hand side is an infinite continued fraction for a0. For the present such an infinite continued fraction simply expresses the fact that ao, ai, a2,... are the successive "incomplete denominators" of o?o as given by the algorithm. We note that if the continued fraction for a0 is finite and ends with an, where n ^ 1, then an ^2 because an > 1. 2. The convergents of the continued fraction for a0.
Assume that either «0 = [ao, ai,..., a n _i, an] or that a0 = [ao,ai, a*,...]. Then define integers Pfc and Qk by the formulae (P-! = 1)
(Po = ao)
jPk = a k P k _i + Pk.2)
(0-1 = 0)'
(Qo = i) ' (Q k =a kQk-i+Qk-2)
ifk
^lj
where k is not greater than n in the first case, but is unrestricted in the second case. p The rational numbers ^ are called the convergents of a0. They are already written as simplified fractions,
(Pk, Qk) = 1,
60
LECTURES ON DIOPHANTINE APPROXIMATIONS
because (1):
Pfc-lQk - PfcQk-1 = (-Dk if k ^ 0. This equation trivially holds f or k = 0 because
P-iQo- PoQ-i = +l. Assume next that k^l and that (1) has already been proved for the suffix k-l. Then Pfc-lQk - pkQk-l = pk-l(akQk-l + Qk-2) - (afcPfc-1 + Pk-2)Qk-l = = - (Pk-2Qk-l - Pk-lQk-2) = - (-I)15"1 = (-l)k, proving the assertion also for the suffix k and so generally. Next, oto may be written as + Pk-2
This formula is certainly true for k=l because 1 a 0 ai+l « 0 - 30 + — = —
—
Assume further that k ^ 2 and that it is already known that _ Pk-2ak-l + Pk-3 °~Qk-2ak-l+Qk-3 ' Since a^-l = afc-l+ — , it follows then that
«o -
Pk-2
-.
-1 +
j-^
+ Pk-3
-
Pk,2
Pk-l«k + Pk-2 9
Qk-2 (ak-1 +—J +Qk-3 (ak-lQk-2 + Qk-3)«k + Qk-2 Qk-l^k + Qk-2 giving the assertion also for the suffix k and so generally. From (1) and (2) it follows in particular that (3)-
a - Pk"1 = (-I)1*"1 ° " Qk-1 " Qk-l(Qk-l«k + Qk-2) '
because p k-l ( p k-l a k +p k-2)Qk-l- p k-l(Qk-l af k+Qk-2) pk-2Qk-l-Qk-lQk-2 a0 - ——= = . Qk s ' Qk-l(Qk-l«k + Qk-2) Qk-l(Qk-l«k + Qk-2) 3. The distinction between rational and irrational numbers. It can now be shown that the continued fractions of rational numbers are finite, those of irrational numbers are infinite. First, every finite continued fraction o?o = [ao, ai,..., 3-n-ljan] has a rational value. For, by (2), applied with k=n and «n=an>
CONTINUED FRACTIONS
61
' 1'""> n" ' nJ Qn-lan + Qn-2 Qn " Conversely, if aQ is a rational number, the continued fraction algorithm for a0 breaks off after finitely many steps. For let the trivial case a0=ao be excluded, and let ao,ai,a2,... and 0i,a2,... be defined as in §1. Then all numbers «k are rational, say — where (pk, q^) = 1 and qk ^ 1. Further r-il = ak-1 K +—,
here fak-l> Pk-l-ak-l 0, Pk-1 - ak-iqk-1 = Qk-l(«k-l - afc-l) ="^" j< q^!
It follows that Pk = Qk-1,
Ok = Pk-1 " ak-iqk_i < qfc-1,
and that therefore qo > qi > ... > 1 . There is then a finite suffix k=n such that Qn=l> and the algorithm terminates with an=an. The result so proved implies that irrational numbers always have infinite continued fractions OiQ = [80,81,81,...].
This continued fraction converges to oto in the sense that (S):
a0 = [ao,ai,a2,...] = lim ^
K—»oo ^K
= lim [ao,aa,...,ak-l,ak]. K— *°°
To prove this assertion, we first note that, by the definition of ak, ak ^ «k < afc + 1,
hence that Qk = Qk-iak+Qk-2 * Qk-iafc-^Qk-2 < Qk-l(ak+D+Qk-2 = Qk+Qk-1 ^ 2Qk . Therefore, from (3), teV '
1 < 1 ^ )„ Pk-1 1 < 1 2Qk-lQk " Qk-l(Qk-«k-l) 1° " Qk-1 1 " Qk-lQk ' (This formula remains valid for rational ot0 provided that k < n.) W>
62
LECTURES ON DIOPHANTINE APPROXIMATIONS Now
Qo = 1, Qi = ai ^ 1, and Qk = akQfc-l + Qk-2 > Qk-1 + 1 if k > 2, so that It follows then from (6), for irrational numbers a0, that 0
as k-*oo,
as was asserted. In fact, even the stronger relation Urn (Qk«o-Pk) = 0 holds because i_
i.l
.1
as
4. Inequalities for |Qk<%-Pk|. For shortness, put 6k = where k is not to exceed n-1 if aQ should be rational. The equation (3) may be written as K
Qk«k+l+Qk-l
'
so that also 6k =
From this equation, 6
k-6k-l
-D +{Qk- (Qk-iafc+Qk-2)}
because «k+l > 1; the case when o?0=ao is an integer must, however, be excluded. Except for this case it has thus been proved that 6k> &k-l> that is, (7):
lQk<*o-Pkl < lQk-iao-Pfc-1 1. In particular, if a0 is irrational, then the numbers
lQk«o-Pfcl form a strictly decreasing sequence tending to zero.
(k = -1,0,1,2,...)
CONTINUED FRACTIONS
63
5. The convergents as best approximations.
Let k^L, and let 2 , where (p, q) =1 and q> 0, be any rational number satisfying the conditions
By the equation (1), there exist two integers a and b such that P « aPfc-1 + bPk, q = aQk-l + bQkHere neither a nor b vanish because then - = g^ or - = • , respectively, I ^CK q ^k— 1 against the hypothesis. It is obvious that a and b cannot both be negative. Nor can both be positive since then q^Qk-l+Qk>Qk against the assumption. Hence a and b have opposite signs. The same is true for the two numbers Qk-lQJo-Pk-1 and Qk«o-Pfc, as follows from the equation (3). Since qao - p = a(Qk-iao -Pfc-l) +b(Qk«o-Pk), this implies the relation (qoio - pl= lallQuflo - Pfc-l' + lb||Qkao-Pkl . Now a and b are integers distinct from zero, so that |a| ^ 1 and |b| ^ 1, and hence |q«o - P\> lQk-i<Xo - Pfc-il+ lQk«o-Pkl> This gives the following result. V (P, q) = 1 and 1 ^ q ^ Qk, then |q«o - pl with equality only if p = Pfc and q = Qk.
The convergents of a0 are thus, in a very strong sense, its best approximations. 6. The rational approximations of g-adic integers.
After this short sketch of the basic properties of continued fractions for real numbers, we proceed to the study of the continued fractions for g-adic and g*-adic numbers. There is no need for dealing separately with the case
64
LECTURES ON DIOPHANTINE APPROXIMATIONS
of p-adic numbers because these may be considered as special cases of g-adic numbers. We begin with the study of g-adic numbers, but, for simplicity, consider only g-adic integers A * 0; thus As was proved in §5 of Chapter 2, such g-adic integers may be defined explicitly in terms of g-adic series
... (g) 0
2
where the coefficients A* ) A^), A< ),... are integers 0, 1, 2,..., g-1. Put Am = A(°) + A«g + A(2)g* + ... + Ato-Dgm-l
(m = 1,2,3,...
so that Am is a rational integer satisfying 0
(9): Here P and Q need not necessarily be relatively prime, but we shall impose the weaker condition that all common prime factors of P and Q are divisors of g. Therefore (P, Q) is a factor of some power of g. We first show that P and Q satisfy (9) if and only if |QA m -P| g «g- m .
(10):
By iQlg^l, this follows immediately from (8) and from the two inequalities lQAm-P|g = I(Q^-P) - Q(A-A m )lg < max(|Q4-P|g, U-A m l g )
and |QA-P| g = |(QA m -P)+QU-A m )| g ^
max(|QAm-P|g, U-Am|g).
Next, the inequality (10) is equivalent to the congruence and hence to the equation
(11): QAm - P where R is a further integer. 7. The continued fraction algorithm for a g-adic integer. The equation (11) may be written as
It suffices to consider those integral solutions P, Q, R for which 0^ |p|
CONTINUED FRACTIONS
65
since new solutions are obtained if multiples of gm are subtracted from P and Q while R is changed suitably. As will become evident further on, it is convenient to allow both signs for P. The algorithm for finding solutions P, Q, R is now as follows. We apply the ordinary continued fraction algorithm to the real number „(«!)_ Am «o - i sr. This number is rational and, by (8), it satisfies the inequality
0 ^ «0(m) < 1. Therefore its development into a continued fraction has the form /1oV (12):
(m) rn (m) (m) (m)-, a = 10, ai , a2 , ...,Q a^ I / \ /_. \ 0 a /___ \ are where Nm, ai , a2 ,•••» Nm certain positive integers. All these integers, and in particular the number Nm+l of terms of this continued fraction, will in general vary with m. With a slight change of notation, let (m) (k = -1,0,1,2,..., Nm) be the convergents of the continued fraction (12); here
=1
= 0 jjj
Qo
=
1
Further put P(km) = AmQtm)- gmR[m)
(13):
(k = -1,0,!,..., Nm).
By the property (1) of the convergents of a continued fraction, (m) n (m) R Q R
(m) (m) / -,k k-l " Qk-lRk = v-1) »
k
so that in the present case Q^' then that
and R^
(Pfc n) , Qk^ ) is far all suffixes Thus P^ Next
and Q^
are relatively prime. It follows k a divisor of gm.
satisfy the condition for P and Q imposed in §6. Q
= 0, 0 < c
« gm
with equality at most when k = Nm because (14):
^m) (m)
(k = 0,1,..., Nm),
66
LECTURES ON DIOPHANTINE APPROXIMATIONS / \ There are also simple inequalities for P ,_ ' . In order to obtain these, t \ ' denote by o^ the real number analogous to a^ that belongs to the continued fraction (12). Then, in particular, so that again
4m) - <& < <4m)
We find now from the equation (3) that (m) afej
ffl
°'
and hence
Qk-1 ^
-
*~
nas the 8i
This equation shows that P^
,
-
^
M
""'
(-l)k and satisfies the inequalities & = 0,1,..., Nm-l).
2Q
k+l
When k=-l or k=Nm, P^
From (13) and (15), Pfe |Q(km) A-Pkm)|g
is given by
and Q^
(16): * g-* We distinguish now two cases, If
satisfy the inequalities
0 < |Pkm)Qkm) l< g» (k = 0,l,2,...,Nm-l).
then there exists a suffix k with 0 < k < N m _x for which, in addition to (16), also m m) ( an) (17): 0<max(|P k |,Q k )* g2 . For; by the construction,
It is then possible to choose a suffix k such that
CONTINUED FRACTIONS
*
67
m
and, for this suffix, by (15), m
whence the assertion. If, however,
m N
m
then it follows from (14) that the greatest common divisor, dm say, of Am and gm is greater than g . Now, by (13), P^1' is divisible by dm; further by (15), m p[m) + 0, hence |p^| £dm > g2 tt 0 *$ k < Nm-l. Thus, in this second case, there can be no suffix k with 0 ^ k < N m _ i for which both (16) and (17) are satisfied. However, now m (18): lO^A-Olg^g-™ 0 < Q J g j < g 2 . By what has just been proved, the algorithm leads, for every positive integer m, to the effective construction of a positive integer Q and of a second integer P such that m m |QA-P|g ^ g- , max(|p|,Q) ^ g2 . The reader will have no difficulty in proving the following result. If A is not a positive integer, and if, in addition, all components of A are distinct from zero, then the pair of inequalities (16) and (17) has solutions for infinitely many distinct values of m. 8. Two numerical examples. The continued fraction algorithm for g-adic numbers has not only some theoretical interest, but is also quite useful for the actual computation of approximations. As a first example, consider the 5-adic number 5 = 2 + 1.5 + 2.52 + 1.5s + 3.54 + 4.55 + ... (5), which is a root of the algebraic equation x2 + 1 = 0.
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LECTURES ON DIOPHANTINE APPROXIMATIONS
We find that the integers Am have the values A! = 2, Aa = 7, A3 = 57, A* = 182, A5 = 2057, Ae = 14557,... and that therefore
«l*> =| «*> -i, <4°> =f , «w = «, «« -3P, «<•> = These rational numbers are equal to the continued fractions, (jL [0,2,2] 2)
= [0, 3, 1, 1, 3],
(3)
<*o = [0, 2, 5, 5, 2], «0W= [0, 3, 2, 3, 3, 2, 3], «oB)= [0» !> *> *> 12» *> *» 12» *> 2l <*W= [0, 1, 13, 1, 1, 1, 2, 2, 1, 1, 1, 14]. From these continued fractions, we immediately obtain the solutions |24| + 7 Is ^5"4,
1 1176 - 44 IB ^ 5"8,
|41£ + 38 18 < 5"5,
of both (16) and (17). It is rather remarkable how small, in comparison with gm, are the values of |p[m)Q^m)|. As a second example, let A^-*-(l,0) be the 6-adic number with the 2-adic component 1 and the 3-adic component 0. It is easily found that A has the 6-adic series
A = 3 + 1.6 + 2.62 + 0.63 + 5.64 + 3.6B + ... (6). Hence Ai = 3, Aa = 9, As = 81, Ai = 81, As = 6561, Ae = 29889,... and therefore
All integers Am are odd and divisible by 3m; by (13), the integers P^ then likewise divisible by 3m and hence
m because 3 > V6*. Thus now the second case of §7 holds, and there are no solutions of both (16) and (17). However, we have now the solution m a
of (18).
m
, 0 < 2 < 62 ,
are
CONTINUED FRACTIONS
69
9. Final remarks to the g-adic algorithm. I believe that the algorithm sketched in the last sections is worthy of a more detailed study, and I have little doubt that many interesting properties will then be discovered. One possible approach arises from the following facts. The numbers ,(m) Am ° g51 occurring in the algorithm are not independent. Since
^
is connected with ct^'
by the relation a 0 ( m ) +A ( m )
m
Here A may assume only the g values 0, 1, 2,..., g-1. There is thus associated with A an infinite sequence of linear transformations {Ti, T2, TS,...} where g This sequence is thus formed by repeating not more than g distinct elements. WJien A is rational, the sequence is periodic; i.e., there are two positive integers m0 and n such that T
m+n = Tm if m ^ m0. One may therefore expect some simple laws relating to one another the continued fractions of consecutive numbers oto and ao + . It further seems probably that there is some non-trivial connection to the theory of the modular group and its congruence subgroups. In a very similar theory for p-adic numbers this was indeed the case as I proved in an earlier paper2. There would be no difficulty in extending the method of that paper to the g-adic case. 10. The continued fraction algorithm for g"-adic numbers.
The continued fraction algorithm for g-adic numbers has an analogue for g*-adic numbers. We shall consider only such g*-adic numbers
2
Annals of Math. 41 (1940), 8-56.
70
LECTURES ON DIOPHANTINE APPROXIMATIONS
which have the property that their g-adic component A is a g-adic integer; for the real component a no restriction is necessary. The component A may again be written as a g-adic series •< A = A(°) + A( 1 )g + A(2)g3 + ... (g), where the coefficients A(m) are integers 0, 1, 2,..., g-1. Just as in §6 put Am = A(°) + A«g + ... + A(m-1>gm-1, so that
U - Am|g < g-m, 0 < Am ^ gm -1. We found then that the integral solutions P, Q of the inequality are identical with the integral solution P, Q of
QAm - P = gmR where R is a further integer. Assume now that such a solution P, Q has the additional property that p Q is a close approximation to the real component a. Then
Qa-P = Qa - (AmQ - gmR) = Q(a - Am) + gmR is small. We therefore put (m)_Am-a 0° = gm and demand that
•p
(m}
is close to 0o . This leads to the following algorithm.
Develop the real number |3o
into a continued fraction (m) r (m) (m) (m) -, 0o = [bo , bi , ba ,...J.
,Here bo ba
is an integer which may be positive, negative, or zero, and bi
,
, bs
,... are positive integers. The continued fraction terminates if / \ and only if a and hence also 0o is rational. R.(m) As in the g-adic case, denote by \ the convergents of this continued
fraction so that
I?' = 0
Q<m) = l
Q(m) k
for k = 1,2,3,...
Further put again = AmQ(km)-gB>R(km)
(k = -1,0,1,...).
CONTINUED FRACTIONS
71
Then, as before, (P[m), Q^) to « divisor of gm because (Q(km), R(km)) = 1. The construction implies that
lQ ( k^-P ( k m) lg*g- m . Further, by the equation (3), (m) 0 (m)/ 0Q Q
(m) o(m k-l < k-lP k + Q
where j3k is the real number analogous to the former number o?k that belongs to the continued fraction. This equation may be written as
In this equation,
.(m) . Jm). , (m) so that, similarly as before, (m) n(m) . n(m) n(m) n (m) n(m) Qk ^ Qk-1 ^k + Qk-2 < Qk + Qk-l ^ Hence, on changing from k to k+1, it follows that
9n(m) 2Q
k
•
Exclude the case when a is rational, so that this inequality is valid for all k?0. Since there exists then for every positive integer m a suffix k such that (m) . 2m fAm) Qk < g * Qk+1 » and for this suffix, both |Q[mWkm)|* g-m and |Q ( k m) A-P ( k m) | g ^ Now the g*-adic value of any g*-adic number J3*— (]8,5) was defined by the equation
g-m.
72
LECTURES ON DIOPHANTINE APPROXIMATIONS
The following result has thus been obtained. Let the real component of A*~+*~(a,A) be irrational, and let the g-adic component be a g-adic integer. For every positive integer m, the continued fraction algorithm allows to construct a pair of integers P f c l ) . Q k a ) > 0 such that |QJf >A* - P(km)|g* * g-m, 0 < max(|P(km)|, Q^) * g2m. This result remains true for rational a, as can be shown, but it then takes a rather trivial form. For now
as soon as m is sufficiently large. The remarks made with regard to the g-adic algorithm in §9 may be repeated for the g*-adic algorithm. For also here consecutive numbers j8o ' and j3o + are again related by the transformation Tm.
PART 2 RATIONAL APPROXIMATIONS OF ALGEBRAIC NUMBERS The problem and its history. Let a be a real algebraic number of degree n^2; thus a. is irrational. One of the results obtained in the proof of Theorem 1 of Chapter 3 was as follows. Let F(x) = Aoxm + Aix™-1 + ... + Am * 0 be any polynomial with integral coefficients, of degree at most m, and of height A = |F(x)|=max(lAol, |Aj,..., |A m |) £ 1. Then either F(a) =0 or |F(a) 1 5* Ci (m)^m'l\ where ci(m) > 0 depends on a and on m, but not on A. Let now m=l and F(x)=Qx-'P where Q > 0 and P are integers; then A=max(fp|, Q), and on putting GI = Ci(l), the last result implies that because Qa-P* 0. This inequality is equivalent to (1):
where c > 0 is another constant depending only on a . For either and then
Q,-n
or
^-|<|a|+l, hence max(|p|,Q) < (|a |+1)Q, and then
'-tl't The inequality (1) is due to J. Liouville1 who used it in his construction of real transcendental numbers. Apart from the value of the constant c, it is best possible for quadratic irrationals (n=2). For, as was proved in two different ways in Chapters 3 and 4, if a is any irrational number (not necessarily algebraic), then there are infinitely many distinct rational numbers p T? such that 1. C. B. Aoad. Sci. (Paris), 18 (1844), 883-885, 910-911.
73
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LECTURES ON DIOPHANTINE APPROXIMATIONS
(2):
-
Let, however, a be a real algebraic number of degree n^3. Then the inequality (1) can be improved. Denote by M(a) the least upper bound of all positive numbers p for which the inequality
has infinitely many distinct rational solutions Q. From the inequalities (1) and (2) it is evident that
2 < M(a) < n. The first improvement of the upper bound for M(ot) was obtained by A. Thue2 who showed that
M(a) * | + 1. Not only was his work of great importance by its implications for the theory of DiophanUne equations, but, in addition, the method introduced by him formed the basis for all later work on the subject. Next, C. L. Siegel3 proved that
an inequality which is of great importance in the theory of Diophantine equations. A further improvement was given by F. J. Dyson4 who found that
M(a) a result also obtained by A. Gelfond. Finally, K. F. Roth5 has settled the problem by proving that M(a) = 2. This result may be stated in several equivalent forms. It implies that for p >2 the inequality (3) has at most finitely many rational solutions and hence that there is then a constant y (a,p) > 0 such that PI P a-Q Py(a,p)Q~P for all rational numbers Q .
I
However, no method is known for actually finding such a constant y (a,p), a disadvantage shared by the methods of Thue, Siegel, Dyson, and Roth, and also by the work in this second part.
2. Skrifter udgivne af Videnskabs-Selskabet i Christiania, 1908, and J. reine angew. Math. 135 (1909), 284-305. 3. Math. Z. 10 (1921), 173-213. 4. Acta math. 79 (1947), 225-240. 5. Mathematika 2 (1955), 1-20 and 168.
APPROXIMATIONS OF ALGEBRAIC NUMBERS
75
Before Roth, Th. Schneider6 had already proved a weaker theorem. Assume there exists an infinite sequence of rational numbers
!?•§•'•!•' •••' where i * QI < ^ < Qs < •••'
such that (4):
|a-^|*Q^
(k = 1,2,3,...)
for some p >2. Then
This theorem by Schneider is nearly as powerful as Roth's theorem for certain applications to proofs of transcendency. Generalisations of these results by Siegel and Schneider have been known for many years. Already Siegel himself8 extended his result to the approximations of a by the numbers of an arbitrary algebraic number field of finite degree. The corresponding analogue of Roth's theorem has recently been established by W. J. LeVeque7 . As these lectures do not deal with the p-adic completions of algebraic number fields, such generalisations will not be discussed. But it would have much interest to carry out a similar extension of the later Approximation Theorems. See, however, Appendix C. In another kind of generalisation, the numerator P or the denominator Q p of the rational approximation •=- is restricted by some arithmetic condition. For instance, it may be demanded that Q is a power of a given positive integer or that, more generally, the greatest prime factor of Q is bounded. Such theorems were given by Schneider6 and myself8, but asserted only a result of the form (5). However, now that Roth's method is known, D. Ridout9 has obtained an extension of this kind for Roth's theorem which is free of this defect. A third kind of generalisation will seem natural to the reader of the first part. Instead of studying the rational approximations of a real algebraic number, one considers those of a p-adic, g-adic, or g*-adic algebraic number a. In the notation of Chapter 3, it is then especially the behaviour of the function
which is of interest. Some 25 years ago, I10 studied exactly this kind of problem by means of Siegel's methods. Again Ridout11 has obtained the analogous extension of Roth's theorem. 6. J. reine angew. Math. 175 (1936), 182-192. 7. Topics in number theory, vol. 2, chapter 4 (Reading, Mass. 1956). 8. Proc. Kon. Akad. Amsterdam 39 (1936), 633-644, 729-737; Acta Arithmetica 3 (1938), 89-93. 9. Mathematika 4 (1957), 125-131. 10. Math. Ann. 107 (1933), 691-730; 108 (1933, 37-55). My results have been extended to the approximations of p-adic algebraic numbers by C. J. Parry, Acta math. 83 (1950), 1-100. 11. Mathematika 5 (1958), 40-48.
76
LECTURES ON DIOPHANTINE APPROXIMATIONS
The aim of the following chapters may now be stated as follows. We shall combine the method of Roth with the idea of Schneider on arithmetic restrictions for P and Q and that of mine on the use of p-adic algebraic numbers. By deliberately applying g-adic numbers and the g-adic pseudovaluation, it will be possible to simplify many of the proofs, as compared with my old paper10. Although the following proofs will make essential use of both real and g-adic numbers, at least one form of the final results will be completely free of these numbers and state a property of rational numbers only. Thus real and g-adic numbers will serve as tools, but not as an end in themselves. This seems to me highly satisfactory. For the theory of numbers still has its main interest in what it can tell us about the rational numbers and the rational integers. But if we want to find properties of the rational numbers, nothing must stop us in the choice of methods used for this purpose.
Chapter 5 ROTH'S LEMMA 1. Introduction Roth bases the proof of his theorem on a general property of polynomials which is to be proved in this chapter. This property is roughly as follows. Let A( Xl ,...,x m )=
2 ii=0
im=0
be a polynomial in m variables, with integral coefficients which are not "too large" in absolute values. Assume that
is a "very small" positive number. Further let
_ PI Qi ' *"'
Pm Qm
be m rational numbers written in their simplified forms for which both the maxima Hi =max(|Pi|,|Qi|),..., H m = max(|P m l, iQml) and the quotients logH2 logH3 logHm log Hi' logH2''"' logHm-1 are "very large". Then A(XI,..., xm) cannot vanish to a "very high" order at xi = KI ,..., xm=Km. (An exact formulation of Roth's Lemma will be given at the end of this chapter). The main idea of the proof consists in an induction for m, the number of variables, the case m=l being trivial. This induction uses a test for linear independence of polynomials in terms of the so-called generalised Wronski determinants. 2. Linear dependence and independence. Let
fy = fj/(xi,..., xm)
(v = 1,2,..., n)
be n rational functions of m variables, with coefficients in a field K. The functions are said to be linearly dependent (or for short, dependent) over K if there are elements ci,..., cn of K not all zero such that 77
78
LECTURES ON DIOPHANTINE APPROXIMATIONS
Cifi+... + cnfn 5 0 identically in Xi ,,.., xm. If no such elements exist, then the functions are called linearly independent (or for short, independent) over K. Evidently, if fi ,..., fn are independent, none of these junctions can vanish identically. Assume, in particular, that the coefficients of fi ,..., fn lie in the rational field F , and that these functions are dependent over the real field P. Then the functions are also dependent over r. For the identity cifi +...+cnfn = 0 is equivalent to a finite system of linear equations ci^ior + ... + cntoia = 0 (a = 1,2,..., s) for GI,..., cn with rational coefficients $va. By the hypothesis the rank of the matrix of this system of equations is smaller than n. The system has therefore also a solution ci,..., cn in rational numbers not all zero, whence the assertion. Conversely, if fi,..., fn have rational coefficients and are independent over F, then they are also independent over P. 3. Generalised Wronski determinants. The letter D, with or without suffixes, will be used to denote differential operators of the form
8Ji + ... + Jm where ]i,..., jm are non-negative integers. The sum ji+...+jm of these integers is called the order of D. Thus the unit operator 1 has the order 0 because ji=...=jm=0« Let i,-.., xm) (v = 1,2,..., n) be n rational functions with real coefficients, and let Di,..., Dn* be n differential operators such that the order of Dv does not exceed v-l (v = 1,2,..., n). The determinant Difi Dif 2 ... D^ / D 2 f a ... D 2 f n \ fi ... Dnfi Dnf2 ... Dnfn is called a generalised Wronski determinant or a Wronskian. This Wronskian evidently vanishes identically when the operators Di,..., Dn are not all distinct. It also vanishes identically if fi,..., fn are linearly dependent over the real field. For an identity cifi +...+ cnfn = 0 implies the n identities
ROTH'S LEMMA .. ... + cnDyfn
79 E
,,..., n).. 0 (v = 1,2,..., If now ci ,..., en are not all zero, then the determinant of this system of linear equations for Ci ,..., cn vanishes, and this determinant is the Wronskian we considering. Let these two trivial cases be excluded. It will then be proved that at leastt one Wronskian of the given functions is not identically zero, at least when fi ,..., fn are polynomials. 4. The case of functions of one variable.
Let
fi, = Mx) (v = 1,2,..., n) be n rational functions in one variable x which have real coefficients and are independent over the real field; thus, in particular,
fn(x) * 0. There is only one Wronskian of these functions that does not vanish trivially, viz. that Wronskian which belongs to the operators d2
d
dn-1
We show by induction for n that this Wronskian is in fact distinct from zero. This is obvious for n=l since then
to. Let therefore n ^ 2, and assume that the assertion has already been proved for n-1 functions. Put
These n-1 functions are still independent. For any equation ClF1+...+Cn-lFn-l
-
, with real coefficients implies, on integrating, that cifi+...+c n -lfn-l B where cn is a further real number, whence ci = ...=cn-l=Cn=0 because fi ,..., fn are independent by hypothesis. It follows then from the induction hypothesis that
(?;:::£:!)
*0
Next one easily shows that, for any rational function g, identically
)•&*(
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LECTURES ON DIOPHANTINE APPROXIMATIONS
Here choose g=fn~1. Then in the Wronskian on the left-hand side all but the first element of the n-th column vanish, and this determinant reduces to \jug... ins/
\*i-« An-i/
* 0.
Hence, finally, /^Di... Dm _ I DI ... DH-!\f~tt ± n
V fi ... fny
\Fi... Fn-iy
'
whence the assertion. 5. The general case.
From now on let
ri rm /\ fp(xi,..., xm) = L ••• Li *ii—im Xll -« xm ii=0 im=0
(^ = 1,2,..., n)
be n polynomials in xi,..., xm that have real coefficients and are independent over the real field. We want to show that at lea'st one of the Wronskians in these functions is not identically zero. In the special case m=l this assertion has just been proved, even for the more general class of rational functions. To reduce the general case to this special one, denote by x a new variable, by g a positive integer exceeding all the degrees ri,..., rm, and put YI
ffn
U=0
im=0
1
(v = 1,2,..., n). ?
m
1
The exponents ii+iag+i3g +...+img "" .of x maybe considered as representations to the basis g, with ii, i2,..., im as the digits; for by the choice of g these numbers may assume only the values 0, 1, 2,..., g-1. Since there is only one representation of any integer to the basis g, it follows that no two terms in the multiple sum for ^(x) are constant multiples of the same power of x. This implies that <£i (x),...,
But then cifi+...+c n fn E 0> whence ci=...=cn=0 by the assumed independence of fi,..., fn« The result of §4 may then be applied to fa,...,
d2 *n
'*'•
ROTH'S LEMMA
81
Denote now by 6 the linear operator
5
-
and by the sign ( ) * the operation of substituting x, xS, xS*,..., xS111"1 for xi, X2, xs,..., xm, respectively. In this notation, evidently and It follows that repeated differentiation of <£j,(x) leads to a relation 1, . . . ,xm)}*.
Here N^ is a positive integer depending on f-m/^1,..., i/'/zNn a1"6 polynomials in x; and Dpii,..., d^Nn arp differential operators at most of order p in the variables xi ,..., xm. The polynomials and the operators depend on ju, but not on the functions fy or fo/. Replace now in the determinant W(x) the terms -=-» ^(x) by their expressions in the derivatives of fj/(x). Then each term in the determinant becomes a sum, and W(x) takes the form Ni Nn-1
Since W(x)^ 0, at least one of the terms on the right-hand side likewise is not identically zero, so that, say, \
But then also
(J
D
^ ^
... Dn lkn.A ^0
\ii i2 is ... We have thus the following result1
In
/
Lemma 1: Let fi,..., fn be n polynomials in m variables that have real coefficients and are independent over the real field. Then there is at least one Wronskian ( * *" - n ) that does not vanish identically. The \ii... in / same assertion holds if the polynomials have rational coefficients and are independent over the rational field. The second assertion of the lemma holds, of course, because, as we saw in §2, the polynomials are also independent over the real field. 1. For a detailed study of the generalised Wronskian see, in particular, A. Ostrowski, Math. Z. 4 (1919), 223-230.
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LECTURES ON DIOPHANTINE APPROXIMATIONS
6. An Identity. By means of Lemma 1 we shall prove a general identity which is basic for the later induction. Assume that m ^2, and denote by ri rm . A(XI ,...,xm) = Z — S aii ...im xl1 — x^ * 0 ii=0 im=0 a polynomial with integral coefficient. Since A(X1 ..... xm) =
im=0\ii=0 im_i=0 it is always possible to write the polynomial in at least one way as a sum n A(XI ,...,xm) = £ -Pi/xi i— >xm-i)Sj,(xm) v=l where Pi,...,Pn are polynomials in xi,...,xm-i and Si ,...,2^ are polynomials in xm, all with rational coefficients. From now on choose one such representation for which the number n of terms is a minimum; then Let us call this the minimum representation of A. In the minimum representation, both the n polynomials Pi (xi ,...,xm-l),.-., Pn(xi ,..., and the n polynomials are independent over the rational and hence also over the real field (§2). For assume, say, that there are rational numbers ci ,..., cn not all zero such that ciPx +...+cnPns°J let e-g-> cn+0. On solving for Pn, Pn s yiPi+—+yn-lPn-l where yi,...,yn-l are rational numbers. Hence we obtain a new representation of A, n-1 ^ # A(xl9...,xm) = 2) Pl/-l(x1,...,xm-i)S (xm) where S (xm) = 2v(xm) + rvZn(xm. v=l , with at most n-1 terms, contrary to the definition of the minimum representation. The independence of 1^ ,...,£n is proved in the same way. By Lemma 1 there exist then two Wronskians
that do not vanish identically. Here, in the Wronskian U*,
ROTH'S LEMMA
83
*
with certain non-negative integers jpi,..., Jy m -l On the other hand, in the Wronskian V**, Ji/m D where * = i km^-l
G> = 1,2,..., n).
Denote by D^ and A^j, the new operators * ** 9 ll 3 - xm-l m and
Further put
D12A ... DmA D2iA D22A ... D2nA ... DnnA ••• AmA
A21A A22A ... A2nA An2A ... AnnA Thus W(xi ,...,xm) = C W*(xi ,...,xm)
where C=K) is a certain rational number. On differentiating the minimum representation of A, we obtain the system of identities n Therefore, by the multiplication law for determinants, W*(xi,...,xm) = U*(xi,...,xm-i)V**(xm) and hence also = CU*(xi,...,xm_i)V**(xm). It is obvious that all three determinants U*, V**, and W are polynomials with rational coefficients in some or all of the variables xi,..., xm. Moreover, the stronger result holds that W has integral coefficients. For if ji,..., jm are arbitrary non-negative integers, the partial derivative
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LECTURES ON DIOPHANTINE APPROXIMATIONS
has the explicit form 11ii -ji ... xim-Jm m
and hence is a polynomial with integral coefficients. On the other hand, the general element in the determinant W is exactly hence is such a polynomial, and so the same is true for W. Now a well-known theorem due to Gauss states that if f and g are polynomials in any number of variables with rational coefficients such that the product fg has integral coefficients, then there exists a rational number c±0 such that both cf and c"1g have integral coefficients. On applying this theorem to the two polynomials CU* and V**, we find that there are two rational numbers jx+0 and w¥Q such that U(xi,...,xm-i) = uU*(xi,...,xm-i) and V(xm) = vV**(xm) have integral coefficients, and that further W(xi,...,xm) = U(xi,...,xm-i)V(xm). The following result has thus been obtained. Lemma 2: Let ri
r
5f
ii=0 im=0 be a polynomial with integral coefficients. There exist a positive integer n not greater than rm+1 and a system of n2 operators A
]LB/=-
where j^i,..., jpim-l* Ji/m are non-negative integers such that and that the following properties hold. The determinant AH, A Ai 2 A ... Am A A 2 iA A 22 A ... A 2n A A
mA ^A
... AnnA
does not vanish identically, is a polynomial with integral coefficients, and can be written as a product = U(xi,...,xm-i)V(xm) where U and V are likewise polynomials with integral coefficients.
ROTH'S LEMMA
85
7. Majorants for U,V and W. If 1 1 ^ v? A(xi,...,xm) = E ... E ai ii=0 im=0
and
ri rm B( Xl ,...,x m )= E - E ^ ii=0 im=0
are two polynomials with real coefficients such that 'aii ...im I * bii ...im
for
all suffixes ii ,..., im,
then B is said to be a majorant or majoriser of A, and we write A«B. It is obvious that this relation has the following properties. // A « B and B «C, then A «C. If A « Band C «D, then A*C «B +D and AC «BD. If A < < B, and c is any real number, then cA« |c |B. The relation A«B may be differentiated arbitrarily often with respect to any of the variables. We also use the notation,
and call [A] the height of A. This agrees with the definition of the height of a polynomial in a single variable given in Chapter 3. We consider now again the polynomial ri rm . A(xi,...,xm) = Z ••• E aii...imXl1— *r& ii=0 im=0 of Lemma 2 and denote its height by a= |Al. By the binomial theorem, Hence A has the majorant A(xi ,..., On differentiating this formula repeatedly, we find that
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LECTURES ON DIOPHANTINE APPROXIMATIONS
Here
so that In particular, it follows that AMy A<<2ri+"'+rma(l+xi)ri... (l+xm)rm Now, by its definition as a determinant,
(&v = 1,2,..., n).
where the summation extends over all n! systems of suffixes Mi>JU2,... that are permutations of 1,2, ...,n. Therefore, on replacing the factors AJUJ/A by their major ants, )ri ... (l+xm)rm}n. Since U+x)nr=Z k=0
«2nr(l+x+...+xnr),
this formula may be simplified to W(x1,...,xm)«22n(ri+-+rm)ann The majorant for W so obtained implies analogous majorants ,...,^.!) «22n(ri+-+rm)ann! V(xm) «22n(ri+-+rm)ann! for U and V. For, by construction, W=UV where U depends only on the variables xi,...,x m _i and V only on the remaining variable xm; furthermore, all these polynomials have integral coefficients. Thus the product of any coefficient of U with any coefficient of V is a coefficient of W. Since the non- vanishing coefficients have at least the absolute value 1, it follows that max(ITjl, [Vl) < Iwl, whence the asserted majorants for U and V. In this way the following result has been proved. Lemma 3: Let A, U, V, W, and n be as in Lemma 2, and let a=2 2n(n+... + r m ) n ,
[^n. ^ nr1,...,pm.i=nrm-i,pm = nrm.
ROTH'S LEMMA
87
Then max([ul, [Vl, [ W | ) « a , and the degrees of U in xi,...,xm_i do not exceed pi,...,pm-l> && degree of V in xm does not exceed pm, and the degrees of W in xi,...,xm do not exceed pi,...,pm> respectively. 8. The index of a polynomial. For any m£l, let
be m rational numbers written in their reduced forms so that (P1,Q1) = ... = (Pm,Qm) = l. The positive integers Hi =max(|Pil,|Qi|),...,H m = max(|P m |,|Q m |) are then called the heights of KI,..., Km, respectively. Let further pi,...,pm be m arbitrary positive numbers, and let again rm ^ i i A(xi,...,x m )= Jj — E ai i m xi 1 ...x n m *0 ii=0 im=0 be a polynomial in xi ,...,xm with integral coefficients which is not identically zero. Hence the derivatives x ) 3 il+ - +3m A(x 1 ..... xm) »--> x m' ~ * 4 jil... Jm!9x3i1...3xmm cannot all vanish at the point Xi = KI ,...,xm= /cm. Denote by J(A) = J(A; pi,... 9 pm;Ki y ... 9 K m ) the smallest value of
JL^.. + JaL
Pi Pm for all systems of suffixes ji ,..., jm for which A
Ji— Jm^ 1 '—'^ +°» and put J(A)=« in the excluded case of the polynomial A = 0. The function J(A) of A so defined is called the index of A at the point (/Ci,...,«m) relative to pi,,">PmThis index may also be obtained as follows. By Taylor's formula,
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LECTURES ON DIOPHANTINE APPROXIMATIONS
ri rm + _ + Pm E - Z Ajl...jm(/c1,...,Km)xpl xijl...xm3m, ji=0 jm=0 where x is a further variable. Hence J(A), for A^O, is the exponent of the lowest power of x in this development with a non-zero factor
=
A
3i —Jm^1 »••• From this, it follows at once that if B(xi,...,xm) is a second polynomial of the same kind, then (A): J(ATB) £ min{j(A), J(B)}, (B): J(AB) = J(A) + J(B), where the indices are taken at (KI ,..., /cm) relative to pi ,...,pm< obvious that
K is
further
either J(A) £ 0 or J(A) = «>, and thai? J(A) = 0 if and only if A(/d ,...,Km) * 0. We need one further simple property of the index. Let li ,..., 1m be arbitrary non-negative integers, and let Evidently B.. . («!,..., «m) =H1 )...(? m ) A.* .*(Ki,.. 0 Km) Ji--0m \Ji/ \3m/ ji—Jm
where m . m .A m Z 7^ = Z J| - Z ^h=l ph h=l p h h=l p h Since the index cannot be negative, we obtain then the inequality * ji*=ji+U,...,Jm = im+lm and therefore
(C):
J(Ai1 ...i
m
) ^ max(0, J(A) - Z ph ^ )• h=l
From now on the index J(A) will nearly always be taken relative to
2. Put w(A) a e"J
ROTH'S LEMMA
89
9. The upper bound a m(a; Hi,..., Hm; riM.., rm) If Hi,...,H m are m positive integers, we denote by Q(Hi,...,Hm) the set of all systems of m rational numbers Pi
_ Pm"
written in their simplified forms, (Pi,Qi) = (P2,Qa) = ... = (Pm>Qm) = 1, of heights Hh=max(|Phl,lQhl) (h = 1,2,..., m). If further a^l is a real number and ri ,..., rm are positive integers, we denote by R(a; n ,..., rm) the set of all polynomials ri rm . A( Xl ,...,x m )= E -. E ^..^...x^O ii=0 im=0 with integral coefficients which are of height lAl ^ a. Roth's lemma deals with a number 9m defined as follows. Definition: Let a^l, and let ri,...,rm, Hi,...,H m be 2m positive integers. The symbol denotes the least upper bound of J(A; ri ,..., r m ;Ki ,..., Km) extended over all polynomials AeR(A; ri ,..., rm) and over all systems of m rational numbers (Ki,...,K m )eQ(Hi,...,H m )Several simple properties of '8m follow at once from this definition. First, both sets R(a;ri,...,rm) and Q(Hi,...,Hm) are finite. Hence the least upper bound in the definition of 8m is attained, and there exist a polynomial AeR and a set of fractions Ui ,...,/cm)eQ such that 6m(a> ri...,rm; Hi,...,Hm) = J(A; ri^.^rmj^i j.-.^m)Secondly, the set R(a;ri,...,r m ) does not lose elements when a increases; hence 6m is a non-decreasing Junction of a. Third, as we are considering now indices relative to ri ,...,rm> J(A) is equal to a sum m . E -J* where 0 ^ ji ^ n ,..., 0 < jm < rm, h=l r h and it follows at once that 0 < J(A) ^ m and hence that 0
* ® * m-
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LECTURES ON DIOPHANTINE APPROXIMATIONS
10. An upper bound for
@i (a; r; H).
We begin with the study of <9i . Let A(x)= £ ai xi*0 i=0 be a polynomial in one variable, with integral coefficients such that [A! = max(|aol,|ail,..., |a r l) < a. Let the rational number K = -^ be written in its simplest form, thus (P,Q)=1, and assume that Hence /c*0 and P*0, Q*0. Suppose J(A; r; K)> 0. Then K is a zero of A(x), say of the exact order j>0, and A(x) is divisible by (x-/e)J. By Gauss's lemma from §6, A(x) can then be written as A(x) = (Qx-P)te(x) where B(x) is a certain polynomial with integral coefficients. Hence the lowest and the highest non-zero coefficients of A(x) are divisible by PJ and Q3, respectively. It follows that H3 ^ lAl ^ a, hence that
a result valid also when J(A; r; K) = 0. Therefore always (0i): 11. The property
0l(a r H)
''
^Tlift
if H
*2'
r^.
The induction proof of Roth's lemma in the next sections becomes simpler if the following notation is used. Denote by t a constant such that 0< t*s 1. If b^l is a real number, and SI,...,SM> KI,...,KM are Positive integers, the ordered system of numbers b, si,...^]^, KI,...,KM is said to have the property TM V either, (i) M = l,
ROTH'S LEMMA
91
or, (ii), simultaneously M**2 and
Kh > Si logKi
(h = 1,2,..., M);
isit b ^ KiM Therefore the following inequalities also hold, M h=l By way of application, let m^2; let the ordered system of numbers a, ri ,..., rm, Hi ,..., Hm have the property Fm; let n be an integer such that 1 ^ n ^ rm + 1; and let i m b =2 — n!a n ; Pi,= nri, P2= nr2,...,pm-i = Then the new ordered system of numbers b, pi,.-«»Pm-l> Hi,...,H m _i has the property r m _i. Proof: The first inequalities
phlog Hh ^pilogH! (h = l,2,...,m-l are for m^3 immediate consequences of the assumption that
(h = l,2,...,m). Next, by hypothesis, w ^ o Y-(m-l)m(2nn-l) Hi ^2 ,
whence, trivially, also
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LECTURES ON DIOPHANTINE APPROXIMATIONS
Finally, n « r m + l ^ r i + l « 2ri, hence n! ^ nn ^ 2nri = 2P1, and, by assumption, a«Him
T
f
, ri + r 2 + • • • + r m
Therefore,
D ^ z92n-mri -^pi «an ^ & (2m+l)p1 because
1 , 1 .
1
12. A recursive inequality for &m. I. Let again t be a constant such that 0
A(X! ,..., xm) eR(a; TI ,..., rm) and a set of fractions
(KI ,.-, Km) e Q(Hi ,..., Hm) such that ^m( a > r i>-«-» r m>Hi,...,H m ) = J(A;ri,...,rm; Ki,...,/cm). Denote by n the integer with Kn and by U(xi,...,xm-l), and W(xi ,...,xm) the three polynomials that correspond to A by Lemmas 2 and 3, and put again b =2 ' ' n ! a11;Pl = nn,...,pm-i = nr m _i, p m = nrm. As has just been proved, the ordered system of numbers b,pi,...,pm_i, Hi,...,Hm-i has then the property rm-i. From the construction and from Lemma 3, 1X1 < a; ful < a, IvI ^ a, (wl ^ a, where
ROTH'S LEMMA
93
Therefore the upper bounds for the degrees of U, V, and W imply that U(xi,...,xm-i)eR(b;pi,...,pm-l), V(xm)eR(b;pm), W(xi,.,.,x m )eR(b;pi,...,p m ). Hence, in particular, with the same fractions KI ,..., K m -l> Km as above, J(V;pm;«m) * Si to Pm;Hm). From the identity W(xi,...,xm) = U(xi,...,xm-i)V(xm) and from the multiplicative property (B) of the index, it follows that J(W;pi,...,pm; i»...> ffm) = J(U;pi l ...,pm-i;«i l ,..,jem-l) + J(V*;pm;«m), or
where, for shortness, $m = ®m-lft>;pi v*Pm-i;Hi,...,H m -i) H- 81 (b;pm;Hm). Instead, we may also write because ph = nrh for all h, and so, by the definition of the index, J(W;ri ,..., rm; KI ,..., fm) = n J(W;pi ,...,pm; «i ,-•> «m)Since from now on only indices of polynomials at the fixed point (KI ,...,/tm) relative to the fixed integers ri ,..., rm will occur, we shall write for these indices simply J(W), J(A), etc. 13. A recursive Inequality for
®m. II.
In the inequality
J(W) ^ n$m just proved, we can give a lower bound for J(W) in terms of J(A). For, as in §7, AA-.-A
A,
where the systems of suffixes /*!,..., jnn run over all n! permutations of 1,2,. ..,n, while the operators Ay are of the form
and the j's are non-negative integers such that 3Ml+...+j]Lim-l « M-l,
3,/m = v-1
(lJ>,v = 1,2,. ..,n).
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LECTURES ON DIOPHANTINE APPROXIMATIONS
Therefore A^A = Aj/Jll...]1Lim_1jj;m,
so that property (C) of the index implies the inequality J(A^A) > max(o, J(A) - *£ -j \ h=l h But and so m 1
\
" h=l
^ maxfO, J(A) - **" - ^Therefore, finally, by the properties (A) and (B) of the index, max (o, J(A)-t-^ ). r \ m/
This inequality can be simplified. For shortness put N = [{ J(A)-t}rm] + 1, where [x] denotes as usual the integral part of x. Hence N-l«{j(A)-t}r m N. The case n^ N. Evidently n-KN-1 <{j(A)-t}rm and hence
Therefore n
v=l
whence (1):
j(A)«t+
J(W)
ifn«N.
ROTH'S LEMMA
95
The case n>N. Assume, for the moment, that J(A)s*t and thus N £ 1. Then maxu), J(A)-t-
rm/
0
if
and it follows that N
J(W)
j(A)-t-^i =N{J(A)-1 >-t}-f{j(A)-t} =£ & &
On solving this inequality for J(A), we find that
(2):
J(A) < t +t/— ? r m J(W) < t + 2 t/f n J(W) if n > N,
because « ^
, 1 rm ^ rm . n ^ 1 n n
In the case J(A)N the same estimate J(A) < t + 2 max(j J(W),yi holds. Since J(W)^n*m, it follows that always In this inequality, A was chosen such that J(A) = J(A;ri,...,rm;Ki,.'vKm) = ®m(a-;ri,.. and $m had the value ®m= ®m-l(b;pi,...,Pm-i;Hi,...,H m -i)+ ®i(b;p m ;H m ). We apply now the inequality ( @i) of §10 which shows that
By definition, pi = nri and pm = nrm, and also r m log Hm ^ r i log HI , hence pm log Hm ^ pi log Hi. It was further shown in §11 that , ^TIT P^
96
LECTURES ON DIOPHANTINE APPROXIMATIONS
Therefore
because The long proof of the last sections has thus lead to the following recursive inequality. Lemma 4: Let m^2, and let be any ordered system of numbers tuith the property Tm. Then there exists an ordered system of numbers b; pi ,...,pm-l> Hi ,..., Hm-1 with the property rm-i such that 0m(a; ri ,..., rm; Hx ,..., Hm) « t where
14. Proof of Roth's Lemma.
It is now easy to prove Roth's Lemma: Put cm=2m -3. If the ordered system of numbers has the property rm, then Cmt Proof: We procede by induction for m. First let m=l, hence Hi ^ 2 and a^HF1*. The estimate ( 0i) of §10 implies then that
as asserted. Secondly assume that m^2, and that the assertion has already been proved for all ordered systems of numbers b, Pi v>Pm-l» HI ,..., H m _i with the property rm-i; it suffices to prove that it then is true also for all ordered systems of numbers a,n,...,rm,Hi,...,Hm with the property rm. By this induction hypothesis, the expression ^m in Lemma 4 satisfies the inequality -l(b;Pi,—,Pm-i;H1,...,Hm-i) +t < cm-it2 "(m"2)
ROTH'S LEMMA
97
and therefore Lemma 4 implies that 0m(a;ri,...,rm;H1,...,Hm) ^ t Now 0 + l2-(">-2> )
whence the assertion. We conclude this chapter by stating Roth's Lemma in a slightly weaker, but more convenient explicit form, as follows. Theorem 1: Let 0
Let
be rational numbers such that OPh»Qh) = 1» max(|P h |,|Q h |) = Hh (h = 1,2,...,m). Finally let A(xi,...,xm) be a polynomial of the form ri rm . 1 A(xi,...,x m )= ^ ... £ a^...! xi1..^^ ii=0 im=0 which is not identically zero and has integral coefficients such that laij...i m l ^ a for all ii,..., imThen there exist non-negative integers ji,...Jm satisfying V Jh.^ 2 m+l t 2-(m-D " h=l
T»i^ rh
such that A ! . . . ! , . . . , ^ ) =1=0.
Chapter 6 THE APPROXIMATION POLYNOMIAL 1. The aim.
In the present chapter we shall construct a polynomial ri rm A(xi,...,xm)= Z — Z ^...im*1—x m 1 * 0 ii=0 im=0 which has integral coefficients that are not "not large" and which vanishes to a "very high" order at the point xi= 5 ,...,xm = {; here £ is a given algebraic number. The importance of this approximation polynomial will become clear in the next chapters. The con&truction does not involve valuation theory, but it is convenient to admit finite extensions of the rational number field. 2. The powers of an algebraic number.
Here and further on, F(x) = Fox* + F^"1 + ...+ Ff denotes a fixed polynomial with integral coefficients such that f > 1, Fo * 0, Ff * 0 and therefore F(0)*0. We impose the additional condition that F(x) has no multiple factor, hence that F(x) and its derivative F1 (x) are relatively prime. Let & be an arbitrary (abstract) extension field of the rational field F in which F(x) splits into a product of linear factors The f zeros «=Si,..., If of F(x) are thus all distinct and different from zero. We use the abbreviation c = 2max(|F0|,|F1|,..., |Ff|) so that c ^ 2 is an integer. Lemma 1: For every exponent 1=0, 1, 2,... there exist unique integers gft gi(1),».,gf!:i such that 98
THE APPROXIMATION POLYNOMIAL
99
Fj «J, = gP + gPl ^ + ... H-gj^ { J;1 ty, . 1,2,..., f),
(1): (2):
Proof: First, the coefficients g are unique because the Vandermonde determinant
does not vanish. Secondly, the equations (1) hold trivially for 1 < f-1 with g; =F0 and the other coefficients equal to zero. Third, for
and therefore
so that the coefficients are integers. Finally, c1 if 1 « f-1,
gPl ..... Ig^l) if 1 >M whence the inequalities (2). 3. A lemma by Schneider.
The following lemma is essentially due to Th. Schneider1 . The proof is taken from Cassels* book on Diophantine Approximation. In the appendix, an entirely different proof is used to prove a stronger result. Lemma 2: Let n ,...,rm be positive integers, and let & be a positive number. Each of the two systems of inequalities m . 0 ^ ii ^ri,...,0 ^ i m ^ r m , E rh £ *Z«(m-B) h=l 1. J. reine angew. Math. 175 (1936), 182-192.
100
LECTURES ON DIOPHANTINE APPROXIMATIONS and
in ^ 1 0 < ii ^ ri,...,0 ^ im < rm, V L — h=l r h
has at most s solutions in sets of integers (ii ,..., im). Proof: The two systems of inequalities are changed into one- another by the transformation
(ii ,..., im)—> (ri~ii >•••> r m~^m) and so have the same number of solutions. It suffices therefore to consider the first system. The proof is by induction for m. First let m=l. The system has no integral solution if s>l, and it has not more than
such solutions if s < 1; hence the assertion holds in this case. Secondly let m^2, and assume the lemma has already been proved for inequalities in m-1 unknowns. We may assume that because the assertion is trivial otherwise. For fixed i=im> where OM 2j ^7" ^ and so, by the induction hypothesis, has not more than
possibilities. Hence, on putting
er--
m
TI .2 r
the original system has at most
m
THE APPROXIMATION POLYNOMIAL integral solutions (ii,..., i m -l> *)• be shown that
The
101
assertion is therefore proved if it can
cr^ 1. Now
rm
v
s
i^os-i+|L
_i r v m r . 2
i^o
B-i+|L
. i
T
ft/__
V sa < v S = Z/ L — oT"o ^ 2 2.(i_fL.)2 i=0 is=0 i=0 ss
j\
V .-.
s4
s
x ^IT
whence, by s s2 ^ /m-1
2m
/ 4m 2 -4m
4. The construction of A(xi,...,xm). I.
As before, let ri,...,r m be positive integers. Let further a and s be two positive numbers such that (3): a^l, where f is the degree of F(x). A polynomial of the form
s^4fV2m,
ri rm . x B( Xl ,...,x m )= S - E bii...im i "^m ii=0 im=0
is said to be admissible if, (i) its coefficients b^ f _i m may assume only the [a] + l values 0,1,2,..., [a], and further, (ii) b
ii ...im = °
unless
\ (m-s)< E ~^ < \ (m-w).
From Lemma 2, it follows immediately that the condition (ii) demands the vanishing of not more than
of the (ri + l)...(rm + l) coefficients of B. Hence not less than
of the remaining coefficients of B may still run independently over [a] + 1
102
LECTURES ON DIOPHANTINE APPROXIMATIONS
distinct values. It follows then that there are not less than
admissible polynomials. 5. The construction of A(XI ,...,xm). II. As in the last chapter, put ii
Jl J
1
v
- m
-9il+"'+lmB(xi....,xm)
* '"•'
m/ --
-* IT
i™
Ji!...Jmia*3l...ad»
Then Jl
'"Jm
'
i1=0 im=0
lm m
" \*lJ
\W
has non-negative integral coefficients if B is admissible. The same estimate as in §7 of last chapter leads to the majorant B
Ji ...Jm(xi '-'
Xm)
« a'2ri *"'+rttlO+»i )ri ...
and hence to Here so that B
Ji -Jm^Thus, for all non-negative suffixes ]i,..., jm,
say, 1=0 is a polynomial in one variable x with non-negative integral coefficients fc not greater than .
and of degree not exceeding By Lemma 1, it follows now that
THE APPROXIMATION POLYNOMIAL r1+...+rm , Pi-K»+r m t1 B ( ° Ji...lm V.....V " # #
F
103
(]) ri+...+rm-l (1L* _ * ** "
L
where
*
1=0
Hence
1=0 «
/-\k
k=0 * because |FoM«c. Therefore, for all suffixes ji,...,Jm and 0,
T
Here B^
is an integer since j3J^ and g^ are integers. Each number B^
has then at most 2[2a(4c)ri+-+rm] + i * 5a(4c)ri+"'+rm possible values, and the set of all f coefficients
of
{5a(4c)ri+-+rm}f possibilities. Let (ji,..., jm) run over all systems of integers satisfying m
jh < 1
by Lemma 2, there are not more than
s
"""
4f
such systems. The corresponding set of integral coefficients
104
LECTURES ON DIOPHANTINE APPROXIMATIONS
B*., where 0 = 0,l,...,f-l; 0 < Ji< PI,...,O < ] m
ri+ +rm 5 ri +1 r m
M*= {5a(4c) -
}
-
+1
possibilities. 6. The construction of A(xi,..0xmK HI. There are not less than M=([a] + l) 2 admissible polynomials B. We therefore choose a = 5(4c)ri+-+rm so that
M> M*. There are then more admissible polynomials B than corresponding sets of coefficients B 1 . Hence there exist two distinct admissible polynomials 9 _ ri rm B(Xl,...,xm)= 2 ... ii=0 and ^B(xi,...,xm)= £ ... £ bi^.,1mxl1...xjg1 ii=0 im=0 with the following property: Define integers JW and B'
such that
"' v £ ^
t} ^
=
I 0=0
; f ^ * \ (m-s) ^ n=1 rh
THE APPROXIMATION POLYNOMIAL
105
Put
ri rm Z ••• E ii =0 im=0
Since B and B are distinct, From the construction, the coefficients a^ ^ values not exceeding a, thus satisfying
of A are integers of absolute
|a ij _ im N5(4c) ri+ - +rm . Moreover, 1
...lm
1 2
£?ih . ~. rh n=l
1, 2
'
and furthermore,
m *. * -,^)-0 if ^^l,2,..., Instead, we may also say that AJ^.J^XJ...^) is divisible by F(x) whenever m j, 0^ j i ^ r i , . . . , 0 < ] m ^ r m , 2 -^^ 5 (m-s); h=l r h * for the zeros £1,..., ^f of F(x) are all distinct. We note that the upper bound for the coefficients of A implies again majorants analogous to those found for B. The following result has thus been proved. Theorem 2: Let F(x) = Foxf + Fix1"1 +...+ Ff, where f * 1, F0 =1= 0, Ff * 0, be a polynomial with integral coefficients which has no multiple factors and does not vanish for x=0. Put c = 2max(|F 0 |,|Fi|,..., |Ff|). Let ri,..., rm be positive integers, and let B be a real number not less than 4fV2m. There exists a polynomial ri rm . A( Xl ,...,x m )= E ..- Z aii...imxi1...xmn*0 ii =0 im=0 with the following properties. (1): Its coefficients ailB..im are integers satisfying
106
LECTURES ON DIOPHANTINE APPROXIMATIONS
and they vanish unless
n=i (2): Aj 1>a< j m (x,...,x) is divisible by F(x) whenever m j. . 0 < h < r!,...,0 ^ jm ^ rm, Z rh ^ ^ Z«(m-s) h=l (3): The following majorants hold,
A
Ji...Jm(x'-'x> This theorem will be applied only for large values of m, and s will always be small compared with m. The last two majorants hold, of course, by the formula proved In Chapter 5, §7, since in the present case,
Chapter 7 THE FIRST APPROXIMATION THEOREM 1. The properties Ad, B, and C.
While the last two chapters depended on purely algebraic ideas, we now introduce real and g-adic algebraic numbers and study their rational approximations with respect to the corresponding absolute value or g-adic value, respectively. Here, as usual, g = pf 1 ...pr r *2, where pi ,...,pr are distinct primes, and ei ,..., er are positive integers; the g-adic value \A\g of A**+~(ai,...,ar) is defined by
lo The later occurring gf-adic and g fl -adic values |a|gf and |a|g" are defined analogously. The letter £ always denotes a fixed real algebraic number, and the letter E a fixed g-adic algebraic number. Only £ satisfying and only -a"-— (£i,...,£ r ) satisfying «i+0,-, « r + 0 will be considered. We denote by F(x) = Foxf + Fix*'1 + ... + Ff, where f > 1, F0 + 0, Ff + 0 , a polynomial of lowest degree with integral coefficients having either £, or JET, or both £ and S, as zeros; hence, by Chapter 3, F(x) has no multiple factors. As before, we put c = 2max(|F 0 |, |Fi|,..., |Ff|), so that c > 1. Next we denote by
s= {jcW |JC W |JC W,...} a fixed infinite sequence of distinct rational numbers
(k)
(k)
" where P
% 0 and Q
* 0 are integers such that (P(k),Q(k)) = l.
107
108
LECTURES ON DIOPHANTINE APPROXIMATIONS
We call H = max(|P< k >UQ (k >|) the height of *(k). It is obvious that lim H(k) = oo. k-*» For such sequences E we now define three properties A^, B, and C where d is either 1 or 2 or 3. First, 2 is said to have the property Ad if for d=l: There exist two positive constants p and ci such that
(1):
U ( k ) -Sl^
(Ai):
ClH
(k) p
'
for all k;
for d=2: There exist two positive constants or and c2 such that U(k)-£lg « c2H(k)~a
(A2):
for all k; and
for d=3: There exist four positive constants p, a, ci, and c2 such that (A,): k (k) -||* Cl H (k) - p and U W -ff| g « erf00"* for aU k. The property As includes therefore both properties AI and A2 . If S has the property Ad, then its elements have for d=l and d=3 the real limit £ > and for d=2 and d=3 the g-adic limit or, because CiHfc)-p and c2IT ^"^tend to zero as k tends to infinity. Secondly, £ is said to have the property B if there exist, (i) two integers gf and gft satisfying g' £2, g" £2,
(g',g")=l;
(ii) two real numbers X and ju satisfying
0 < A < 1,
0 < p < 1;
and
(iii) two positive constants c3 and c4, such that (B): |P W | g( *c,H (k)X - 1 and |Q(k)|g» * (^H^'1 for all k. The first inequality (B) holds trivially if A=l as we may simply take c3 = l; and similarly for the second inequality when ju=l. For later it is important to note that if d=2 or d=3, and if S has both properties Ad and B, then
(g, g1) = 1 if 0 ^ A < 1. For lim |P(k)L, =0, while (P(k),Q(k)) = 1, hence |Q(k)Li = 1, and so also k—»°° ^ ° lim l/c(k)L, =0.
k— »oo
f
&
If now g and g had a common prime factor, pi say, then
THE FIRST APPROXIMATION THEOREM lim kl =0 k-->«> Pi
109
a n d l i m U - { i L 1 = 0 , hence £ i = 0 , k-*°o P
contrary to the hypothesis. Third, S is said to have the property C if there exists a positive constant c5 such that (C): U^N c5 for all k. In the two cases d-1 and d=3 the property C follows from the property Ad because
In the remaining case d=2 it is, however, independent of AdOur first aim in this chapter is to prove the following result. Main Lemma: If the sequence has all three properties Ad, B, and C, then r ^ A + p, where rp if d=l, (2): T ~ J or if d=2, LP-KT if d=3. The proof of this lemma will be long and involved, and it will be indirect. It will be assumed that (3): r = A.+/n4e where e > 0, and from this hypothesis we shall deduce a contradiction. 2. The selection of the parameters. Since the property Ad weakens when the exponents p and a are decreased, we may without loss of generality assume that (4):
0 < e ^|.
For the same reason we are allowed to assume that (5): ci & 1, c2 > 1, ca ^ 1, c4 ^ 1, c5 ^ 1. Similar to c, CI,...,CB the letters c fl , c 7 ,...,Ci, C2, C3, TI, T2, and T8 will be used to denote certain positive constants that depend only on the sequence S and the algebraic numbers {, S, or £ and S9 respectively; they will, however, be independent of the numbers m, s, t, Ki,...,K m , ri,..., rm
110
LECTURES ON DIOPHANTINE APPROXIMATIONS
to be defined immediately. The last three constants TI, T2, and T3 will not be fixed until the end of the proof. The parameters are now selected as follows. First, choose a positive integer m such that
and in terms of m define the positive number s by (6):
s=f.
Secondly, choose a number t such that
(7):
1 o-to-D 0 < t * 1, 2m+1 12 « -SL .
Third, select m distinct elements K^ll\ K^,...,K(im) of Z that satisfy certain inequality conditions to be stated at once. To simplify the notation, these elements of S are written as
where the Ph and Qh are Integers for which Ph * 0, Qh + 0, Thus Kb has the height Hh = max(|Phl,|Q h l). The hypothesis of the main lemma imposes, for all suffixes h=l, 2,..., m, the following inequalities:
(Ad):
~Uh-$l'sc1Hnp
ifd=l,
< Uh-Slg^caH^
if d=2,
J Kh~ ? I *CiHhP and I «h-slg«ca Hnff (B):
if d=3;
1
|Phlg' -scaHh" and
(C): It is necessary for the proof to add the following conditions:
(8):
|Ph|g' « g V i f O < X < l
(9):
log Hh+1 * | log Hh
and, depending on the suffix d,
(h= l,2,...,m), (h = 1,2.....m-1),
THE FIRST APPROXIMATION THEOREM
111
/ £m2 £(m-l)m(2m+l) \ Hi £ maxV(20c)t , 2l , Td/ . Since the elements of S satisfy the limit formula (1), it is possible to choose Ki,...,/cm such that all these inequalities are satisfied. Finally, select m positive integers ri,..., rm such that O Irw* TI ri & 21ogH m ri e log Hi ' (10):
(12):
rh ^ ri Jjj|j- > ^
(h = 2,3,..., m) .
Since, by (9) and (10), evidently 2 < Hi < H2 < ... < Hm, these formulae imply that
because, by (4),
Hence we find that (13): rilogHi ^ rhlogH h ^ (l+e)rilog HI (h = 1,2,..., m). Therefore, for arbitrary non -negative exponents ki,..., km, it follows that
We also note that, by (9), (11), and (12), rh * - > 2, rh-1 > 2rb
2rh+1 logHh+l < rilogHi ^ rh l°S Hh,
hence
rh+1 < 2 log Hh < t rh logHh+1 " ' and therefore (15): Thus, trivially, (16): ri > ra >... >rm
r h + i< rht and
(h = 1,2,..., m-1).
ri + r2 + ... + rm <
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LECTURES ON DIOPHANTINE APPROXIMATIONS
3. Application of Theorems 1 and 2. The polynomial !fl(x) has no multiple factors. Hence, by Theorem 2, applied to this polynomial and the numbers m, s, ri ,..., rm, there is a polynomial
ii=0 with the following properties. (17): The coefficients ai^^
... 2 »i1...lm im=0
are integers such that
and they vanish unless
11=1
(18): The derivative Ajx
i
(x,...,x) is divisible by F(x) whenever
0 « ji ^ r!,...,0 < im*rm, t ft ^ |(m-s). h=l rh * (19): We have the following majorants, A
ji...Jm(xi'"->xm) « 5(8c)ri+-+rm(l+xi)ri...(l+xm)rm,
A, lM j m (K f ... f x) « 5(8c)ri+-+rm(l+x)ri+-+rm. By (10), (16) and (17), the height of A(x1,...,xm) does not exceed 5(4c)ri+-+rm < 5(4c)mri < (20c)mri < H?**
.
It follows then from the inequalities (10), (12), and (15) that the hypothesis of Theorem 1 is satisfied for the polynomial A(xi,...,xm) and the numbers m, s, t, KI ,..., /cm, ri ,..., rm. But then, by this theorem, there exist suffixes li,...,lm satisfying the inequalities (20):
O^l^n^O^l^rm, £ h=l
- * 2m+1t2
m
such that the function value
=A
say>
THE FIRST APPROXIMATION THEOREM does not vanish, (21):
113
A(1) * 0.
This number A/,, is rational and so may be written as a quotient D
d)
of two integers N(i) and D(i) satisfying In the next sections we shall establish upper and lower bounds for I Am |. To express these in a simple form, it is convenient to introduce the following abbreviations, m i 1 1 (22): A = £ ik Sl =i( m -s)-A, S2 = i(m+s)-A, S8 = m-A . h=l rh * * It is obvious from the formulae (4), (6), (7), and (20), that
(23):
0 ^ A ^ 3? 1 7
9
1
(24): Si £ j(2-e)m £ ^m,
9 < l
S2 ^ j|(6-e)m £ |gm,
1
9 ^
S3 ^ ^(6-e)m ^
||m.
4. Upper bounds for |A(|)| .
For real xi,...,xm and arbitrary suffixes ji,...,]m it follows from (16) and (19) that
* (40c) mri {(H-lxil)...(l + |x m |)} ri . We apply now the property C of S. This property implies, first, that |A (1) l«ce mri
(25): where, for shortness,
ce = 40c(l+Cs) . For
k !)...(!+ 1 *m I) « (l+c 5 ) m . Secondly, let d=l or d=3. Then, again by property C, m = l i m k ( k ) | * c5 k-*«>
114
LECTURES ON DIOPHANTTNE APPROXIMATIONS
and hence | A4 ...jm(«,...,£) I < cemri
(26):
for all suffixes ji ,..., jm.
The inequality (25) is, of course, valid for all three values of d, but will be used only for d=2. A much stronger upper bound for | A(i) I can be proved in the other two cases d=l and d=3, using (26). From Taylor's formula we obtain the identity ri rm . A(xi,...,xm)= £... E AJ Jm .s (x,...,x)(xi-x);!l...(xm-x):lm, jm=0 and on repeated differentiation, (xi
(27):
'
.0 By putting Xi = KI ,...,xm = Km, x = we find that
(28): A(1) . In this equation,
while, by (18),
.O-O if It follows that it suffices to extend the summation in (28) only over those systems of suffixes (]) = (ji,..., jm) ^at belong to the set J: 0 ^ h-u < n-lif..., 0 ^ jm-lm ^ rm-lm, ^ ^^ h=l rh It is then evident that (29): |A (1) |
A
ri
rm
*• • • " . . . . .
> Si.
THE FIRST APPROXIMATION THEOREM
115
and
A** = max k - {|jl-K.. km-« |Ji»-lm _ Vjj€j
In the first expression,
1=0 so that by (16) and (26), A* * £ •» iT camri • 2il+-+j"a = c?*1 (2ri+1-l) ... (2rm+1-l) < ji=0 jm=0
U h -£|<
Cl Hj;
(h= 1,2,..., m),
which holds also for d=3; here, by (5), GI ^ 1. Hence
A**
(j)eJ
)eJ
m
where
Further, by the left-hand side of (14) and the definition of J,
max (H 1 ' 1 ... (j)eJ
tfm-'P
m
^ max Hl (j)eJ
h=l
so that
We finally put
C7 = 4ci CB and substitute the upper bounds for A* and A** in (29). We so find that (30):
I A(1) | < c7mri Hr^1*
for d=l and d=3.
Here, by (24), the exponent of Hi is negative; hence this upper bound is smaller than that given by (25). However, no explicit use of this fact will be made.
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LECTURES ON DIOPHANTTNE APPROXIMATIONS
5. An upper bound for In the two cases d=2 and d=3 there exists an upper bound for 'very similar to that for I A(i) I which has just been proved. In both cases the sequence S has the g-adic limit S, and so Urn k ( k ) L=|SL. 6 5
k-»oo
There exists then a constant ca > 1 depending only on S such that IK
Ig < c8
for all k,
and therefore also l-5lg« c8. This time we substitute the values in the identity (27), so obtaining the equation ri rm /-\ A \ -i -i (31): A(1) = £ ... £ A1 1 (S,...&fa}J m)(/c1-5)]l"ll...(/cm-5')3m"lm. ji=0 jm=0 Jl"°m w ^lm/ Here, just as in (28), it suffices to extend the summation only over all systems of suffices (j) = (ji,..., jm) in J. The binomial coefficients in (31) are integers, hence their g-adic values are not greater than 1. It follows then from the non-Archimedean property of the g-adic pseudo-valuation that (32): |A (1) | g
and B**
max
k1-5|
The polynomials Aji(.,jm(x,...,x) have integral coefficients and are at most of degree ri+...+rm ^ mri - therefore
and hence also
Next, for the second factor, we apply the property
THE FIRST APPROXIMATION THEOREM
117
which holds also for d=3; here again, by (5), c2 ^ 1. It follows that B** < max C2(3l"ll)+-+(3m"lm). max (Hi!1""11 ...H3™"1^)"*. Here maxc^^-^^m^m) < c ^+-+r m < (j)eJ while, by the left-hand side of (14) and the definition of J,
max (H 1 "" 1 ...H" ... 1""1)^ ^ max H! ! (])eJ (j)eJ
h=l
r
h
< HraSiri
Hence
Put
C9 = C2 CB
and substitute the upper bounds for B* and B** in (32). We so find that (33):
I A(i) |g * c9mriH;aSiri
for d=2 and d=3.
6. An upper bound for | D / 1 1 1 .
In this and the next sections we shall establish an upper bound for I Dm | and a lower bound for |N(i) I ; by combining these, a lower bound for | AQQI will be obtained. From the definition of A(XI ,...,xm),
so that, in particular,
In this equation,
while, by (17),
118
LECTURES ON DIOPHANTTNE APPROXIMATIONS «V..lm-°
UnleSS
l<"i-s)< £ J; < |(m+s). h=l It follows then that the summation in (34) need be extended only over those systems of suffixes (i) = (ii,..,, im) that belong to the set I: Q< ii-1^ ri-li,..., 0 < im-lm « rm-lm, Si < £ h=l Each single term in (34) has a denominator
Therefore the least common denominator D^ satisfies the inequality |D(i)|^ tagql 1 " 11 ...Qfe 1 " 1 " 1 , =D
say,
where the symbol "1cm" stands for the least common multiple. We apply now the second half iQhlg 11 * c4Hh"
Oh = 1,2,..., m)
of the property B. By this property, Qh is divisible by an integral power of g" that is easily proved to be not smaller than
but may be larger. For each suffix h=l,2,.,., m it is then certainly possible to find a factorisation of Qh where Qj* is that integral power of gtf which is defined by the inequalities (35):
^-Hj-M«Q*<
IH^
and where we have put c10 = max^l, — j . The complementary factor Qh* then satisfies the inequality (36):
lQh*l= iQhlQh" 1 **
From the factorisations of the Qh it is obvious that (37): D where
THE FIRST APPROXIMATION THEOREM
and D** = 1cm ( (i)el The first factor D* is the least common multiple of certain integral powers of gu and hence is equal to their maximum, .±
1
^.S
1
D* = Therefore, from (35),
D* ^ max Cin m
"""
• max (Oel
Here Ci0 ^ 1 so that
Further, by the definition of I and by the right-hand side of (14),
m • 1 ir i l1l m (l-]Lt)(l+c)ri £ } max (Hi " ...Hj? " ) "^ < max Hi h=I m (Dd Hence
For the second factor D** we use the trivial estimate
D** Here cig" >1 and hence (C4g")mri. Further, again by the definition of I and by the right-hand side of (14), m (i \ V r (El l^ ...H^-S^ * Hl h=l Thus it follows that
Finally put Cn = C4g'! '
119
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LECTURES ON DIOPHANTINE APPROXIMATIONS
and substitute in (37) the upper bounds for D* and D** just obtained. Since |D(i) |
|D(|)|
7. Lower bounds for |N
We again apply the equation (34) of the last section; by means of it, we shall determine integral powers N* of g1 and N** of g that are divisors of These two powers are relatively prime, so that their product likewise divides N(j). However, in certain cases it becomes necessary to take N* or N** equal to 1, and it may even be convenient to allow lower estimates for these numbers that are smaller than 1. Assume for the moment that 0 < A< 1. Then, by the hypothesis (8), all numerators Pn are divisible by g1, and hence all denominators Qh are prime to g' . The first half IPfalg' ^CsHh"1
(h=l,2,...,m)
of the property B implies that Pn is divisible by an integral power of gf, P* say, which is easily seen to satisfy the inequality
(39):
P£
here c3gf >1. On the other hand, the denominators
of the terms of A(i) and hence also their least common denominator Dfl) is relatively prime to g'.1 It follows that the numerator NQJ of Am is divisible by that power N* of g which is defined by
here the symbol "gcd" denotes the greatest common divisor. All products r
in
1
are, however, integral powers of g . Their greatest common divisor is then equal to their minimum, im lm N* = min P?11"11 ... P* m " .
It follows therefore from (39) that
THE FIRST APPROXIMATION THEOREM
121
Here N ri+...+r m
. -
N
"(shr)
mri
•
Further, by the left-hand side of (14) and by the definition of I, m fi \\* V min (Hi-1' ... HJf-S1-* * min H! > & Therefore, finally,
(40): In the case X=l so far excluded the right-hand side of this inequality does not exceed 1; hence (40) remains valid without the restriction on X. We put
(41):
if d=l.
N** = 1
Next let d=2 or d=3. Then |A(i)|g possesses the upper bound (33). This upper bound implies that N(i) is divisible by an integral power of g, N** say, which satisfies the inequality (42):
N**> ( g - c H r
1
) ' * -E1 \Cog/
i f d = 2 o r d=3.
First assume again that 0 ^X < 1. T
As was shown in §1, g and g are in this case relatively prime, and so the same is true for their powers N** and N*. Hence N*N** is a divisor of N(i), whence This inequality still remains valid for X=l provided N* is then replaced by its lower bound from (40). Therefore, depending on the value of d, a lower bound for |NQ) I is given by the products of either the right-hand sides of (40) and (41), or the righthand sides of (40) and (42). Hence, on introducing the new constant Cl2 = Csgogg',
we arrive at the following lower estimates, for d=l, 1
ciT* Hr~""
lCl 2
rv/M1 1
*
for d=2 or d=3.
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8. Conclusion of the proof of the Main Lemma. Put Cis = CsCiig 1 , CM = CuCis
and Ci = C7ci3,
C2 = c fl c l4 ,
C3 =
The two inequalities (38) for |D(i)| and (43) for |N(i)| immediately lead to aJower bound for
which, naturally, depends on d. The result is as follows:
1
for d=2, or d=3.
On the other hand, the formulae (25) and (30) asserted that for d=1 Qr d=3j
for d=2.
On combining these inequalities, it follows that
c-m
jj
c-m H(
These three formulae may be put into exactly the same form, (44):
Hx E
(d=l,2,3),
where, for shortness, (45): Ed = (l-X+T)S1-(l-]Li)(l-K)S2-M(l«)S3 (d=l, 2, 3), and T denotes the number which was defined in the Main Lemma. We can write the expression for £4 also as + ^e)s-
Here the coefficient of m is equal to
that of -s is equal to
( T -X-e)A.
THE FIRST APPROXIMATION THEOREM
123 2,
and that of -A is equal to 1 r-X-e = jLt+3e ^ 1+3-j < 2;
here we have applied the former assumptions T = X+jLi+4€,
0 < e< j,
0 < JLI < 1.
It follows therefore in all three cases d=l, 2, or 3 that , em o.25_ -
€m
We finally choose the remaining constants Td such that Td > CdT
(d=l,2,3).
Hi £T d
(d=l,2,3),
Since, by (10), it follows that em 3
=C™
(d=l,2,3),
contrary to (44). This proves that the original hypothesis (3) leads to a contradiction and so shows that the Main Lemma is true. 9. The first form of the First Approximation Theorem.
It is now easy to deduce from the main lemma a more general result which we call the First Approximation Theorem. This theorem will be stated in two different forms which, however, are equivalent. The first form of the theorem is nearly identical with the main lemma, except that the condition C is omitted. First Approximation Theorem (I): Let £ * 0 be a real algebraic number and jis—^((;iv>£r)> where £1* 0,...,£r * 0, a g-adic algebraic number. Let p, a, A, ju be real constants satisfying p > 0 , a > 0, O ^ X ^ l , 0 < / * < ! ; let ci, ca, c3, c4 be positive constants,• and let g&2 and g" ^ 2 be fixed integers. Finally let S = {/r1', /P2s «W,...,} be an infinite sequence of distinct rational numbers »
Where P(k)
* °' Q(k) * °'
max
(I
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LECTURES ON DIOPHANTINE APPROXIMATIONS with the following two properties.
(Ad): For all k, U(k)- ll^eiH 0 0 -*
*/d=l,
U (k) -Sl g W k) -*
«/d=2,
U(k)- $| * CiH(k)-P and U(k)-S|g* C2H(k)-" (B): jFor all k,
if d=3.
Then /or d=l, for d=2,
P-KT <X + ju /or d=3. Proof: We mentioned already in §1 that, for d=l and d=3, a sequence S with the properties A& and B has the real limit £ and so possesses the third property C trivially. The assertion is therefore in these two cases contained in the main lemma. There remains then only the case d=2 in which the assertion has yet to be proved. First assume that S contains an infinite subsequence
Si = {/c(il), K(ia)' K(i3),...}, where it < i2 < i3<..., such that, for all k and some positive constant c5, The main lemma may then be applied to Si and gives the assertion. Secondly let £ contain no such subsequence Ei . Then lim |/c(k)| = co, k-*» and hence the sequence of the reciprocals S) So=rfU°U ,..},
where
.fr> .. .W-l . 9J£ ,
• has the property C, (k = 1,2,3,...) for some positive constant Cs. It is obvious that ff and K are of the (k) same height H . Hence So has also the property Bo which is analogous to B, except that X and ju, and also c3 and c4, are interchanged. We finally show that So has the property A2 . By hypothesis, S has this property, 5
THE FIRST APPROXIMATION THEOREM
125
(46): U(k)-S|g < c2H(k)"a (k = 1,2,3,...). Therefore S has the g-adic limit S, and so, for j=l,2,...,r, this sequence has also the pj-adic limits £j. But then the reciprocal sequence So has for all j the pj-adic limits fj1, and hence So has also the g-adic limit Therefore, in particular, Urn Jim k —>oo
and hence there is a positive constant GO such that ko Ig ^ c»
for
^ k«
From (46) and the identity
it follows finally that , where c = c0 I*'1 |g ca. We apply now the main lemma to the sequence So instead of- S and find that a ^ jit + X, giving the assertion. g
10. Polynomials in a field with a valuation. The second form of the First Approximation Theorem makes a statement on the values of a polynomial assumed in a sequence of rational numbers. Before enunciating and proving this theorem, it is necessary to discuss first a property of fields with a valuation. Let K be a field with a valuation w(a), and let Kw again be the completion of K with respect to w. We say that K has the property D if the following compactness condition is satisfied: (D): Every infinite sequence of elements of K that is bounded with respect to w contains an infinite subsequence which is a fundamental sequence with respect to w, hence has a limit in Kw. Let K have this property D, and let F(x) = F0xf+Fixf"1+...+ Ff, where f > 1, F0+ 0, be a polynomial with coefficients in K which has no multiple zero in Kw. Put G(x) = Fo1F(x) = xf+GiXf-1+G2xf-2+...+Gf, y= t+w(Gi)+w(G2)+...+w(Gf), so that
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LECTURES ON DIOPHANTINE APPROXIMATIONS
and also G(x) has no multiple zero in Kw. Assume now that x is an element of Kw such that w(x) > y
and hence
w(x) > 1.
Then w(G1xf-1+Gaxf-2+...+Gf) ^ (y-l)max(w(x)f-1, w(x)f-2,...,w(x), 1) < < (y -Dwtx)*'1, and therefore w(G(x)s>w(x*) - w(Gixf-1+G2xf"2+...+Gf) ^
Conversely, it follows that if
w(F(x)) < w(F0), then w(x) «y,
because the first inequality implies that w(G(x)) = vrCFo1 F(x)) ^ wfFj'MFo) = 1. Consider now an infinite sequence 2 = {ic'1', ic2', K'S',...} of elements of K satisfying lim w(F(/c(k))) = 0. This assumption implies that )
)) < w(F0)
and hence
for all sufficiently large k. Thus the sequence 2) is bounded with respect to w and so, by the property D of K, it contains an infinite subsequence S1 ={/c(il), /c(ia), /c(is),...}, where i1
lim K^ (w), k— »°°
=|
say,1
in Kw. However, polynomials in K[x] are continuous functions with respect to the metric on K defined by w, and therefore F(5) = lim F(K(ik)) = lim F(
THE FIRST APPROXIMATION THEOREM
127
because otherwise £ would be a multiple zero of F(x). From the continuity of the polynomial Fi(x), it follows that lim Fi(K Vk O = Fi(£)*0 (w), and hence that v
) 4s 0 for all sufficiently large k. There is no loss of generality in assuming that this inequality holds for all suffixes k, Hence a positive constant 74 exists such that )) £ yf1
(k = 1,2,3,...).
The equation leads therefore at once to the following result. Lemma 1: Assume that K has the property D. Let F(x) be a polynomial in K[x] which has no multiple zeros in Kw, and let S = {*W, *W, K(*\..} be an infinite sequence in K such that lim w(F(*(k))) = 0. k->°o
There exist an infinite subsequence S1 = {irll% K™\ *risV..} of S, a zero g of F(x) in Kw, ane? a constant n > 0 SMC& ^o^ w(/c(ik)-g) ^ nw(F(fc(ik)))
(k = 1,2,3,...).
11. Two applications of Lemma 1.
In Lemma 1 choose for K the rational field F and for w(a) either the absolute value |a| or any p-adic value |a|p where p is an arbitrary prime. The completion Kw becomes then either the real field, or the p-adic field. Both the real field and every p-adic field have the compactness property D1 . Hence the following two results are contained in Lemma 1. 1
This is a classical theorem in the real case, and it may be proved in the p-adic case as follows. Denote by 2)= {K^, j^W, K($9 . . .} any bounded sequence of p-adic numbers. Let its elements, without loss of generality, be p-adic integers; thus they can be written as series «**= a^ + a P2+. . . (p)
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LECTURES ON DIOPHANTINE APPROXIMATIONS
Lemma 2: Let F(x) be a polynomial with rational coefficients which has no multiple zeros, and let S = {K^' , /r2^, fr3',... } be an infinite sequence of rational numbers such that lim
There exist an infinite subsequence Sf = {tc^\ jc , ic ,...} a real zero | of F(x), and a constant yi > 0, such that k f t k). { | < y i | F ( j c ftk))|
of S,
for all k.
Lemma 2': Le* F(x) be as in Lemma 2, and let S = {«W K® K^, ... } be an infinite sequence of rational numbers such that lim|F(K(k))|Fp = 0. k—»«> There exist an infinite subsequence S" = {*(Jl\ fi*\ $3\...} of S, a p-adic zero £p o/ F(x), and a constant y2 > 0, such that for
^ k-
The second lemma may be extended to g-adic values and g-adic numbers, Lemma 3: Let F(x) be as in Lemma 2, and let S = {K(I] K^, K®,...} be an infinite sequence of rational numbers such that lim I F(*(k)) !„ = (). kThere exist an infinite subsequence S1" = {fc(hl), ic(h2), K(hs),...} of S, a %-adic zero E of F(x), and a constant ys > 0, such that for all k. Proof: From the hypothesis and the definition of the g-adic value, it follows that also
where the digits a^ assume only the values o, 1,. . ., p-1. The set of the first n digits of each KW has thus only pn possibilities. It follows that it is possible to select successively an infinite subsequence ^ = {K^, 1$, 1$, . . .} of L. an infinite subsequence Z, = {$, $, $, . . .} of % an infinite subsequence % = {icty, $, '$, . . .} of Ly etc., such that, for every n, the n first digits of all elements of Zn are identical. The diagonal sequence L'= (tc^, K^\ ^ , . . .} is still a subsequence of L, and it has the property that, for every n, the first n digits of all but finitely many of its elements are identical. Hence 2' is a fundamental sequence, as asserted.
THE FIRST APPROXIMATION THEOREM
129
lim k—>°° We now apply Lemma 21 repeatedly, once for each prime factor p] of g. First, there exist an infinite subsequence Si = {K , K , JT S, api-adiczero £1 of F(x), and a constant y(0 >0, such that
,...} of
for all k. 2i) Secondly, there exist an infinite subsequence S2 = of Si, a p2-adic zero £2 of F(x), and a constant y W > 0, such that
p2
forallk,
I ) |pj
for aU k.
...
while, naturally, also
Continuing in this manner, we obtain for every suffix j=l, 2, ...r an infinite sequence
Sj
= frM, K(^\ fc(hi3),.-}, where
Si 2 S22 ... 2 Sri apj-adiczero gj of F(x), and a constant yj > 0, such that |K (hjk) -*ilpi^ y ft) lF&c (h l k) )lii l Let S
tff
for i=l,2,...,j andfor all k.
be the sequence Sr; further put / logg (1)eilogpl maxU maxyy
logg \ (r)e r logp r / ) , ,..., y
and denote by E the g-adic number S —(?!,...,
|r),
which is algebraic and a zero of F(x). We have then
max
logg \ ^^^lg10^1^
f logg 1 (i) h 1OgPi max ^ (y |F(^ rk))lp> f i=l,2,...,r ^ j for all k
and hence
for all k, whence the assertion.
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12. The property A'd. As earlier in this chapter, let again F(x) = Fox* + Fix*'1 + ... + Ff, where f > 1, F0* 0, Ff * 0, be a polynomial with integral coefficients which does not vanish at x=0 and has no multiple factors, hence also no multiple zeros in any extension field of the rational field. Further denote again by $ a real zero and by S a g-adic zero of F(x), and by p and or two positive constants. Finally let again S = {ft'1', /c'2', /r3',...} be a sequence of distinct rational numbers «W - 5jjJ + 0
of heights
H = maxdP^I, |Q(k>|)
such that P ( k >*0,Q ( k ) +0, (P
is a polynomial in x and y with integral coefficients. Evidently I F(K(k)) | = U«-« | |*(K(k),«) I (k)
W
(k)
I F(/c ) | - | « - S| _ |*(/c , S) | _
'g
if d=l or 3, if d=2 or 3.
THE FIRST APPROXIMATION THEOREM
131
Further, by the hypothesis, K^ has the real limit £ if d=l or 3, (k) KV ' has the g-adic limit S if d=2 or 3. This means that S is a bounded sequence with respect to the absolute or g-adic values, and hence that the numbers l*(jc^,{)| for d=l or 3, and \^k\S)\s for d=2 or 3 are bounded. Let their upper bounds by Fi and r2 , respectively; it follows then that S has the property Ad with the constants ci = ciTi and ci = car2 , respectively. Secondly let S have the property Ajj. If d=l or d=2, the assertion is contained in Lemmas 2 and 3, respectively. If, however, d=3, both lemmas must be applied one after the other. First, by Lemma 2, there is a real zero £ of F(x) and a subsequence Si of S which has the property Ai with respect to £ . Secondly, by Lemma 3, there exists also a g-adic zero S of F(x) and a subsequence Sf of £1 which has the property A2 with respect to S. Since S1 still has the property Ai with respect to £ , it has then the property A3 with respect to both £ and S, whence the assertion. 13. The second form of the First Approximation Theorem.
By combining the lemma just proved with the first form of the First Approximation Theorem we immediately obtain the following second form of the theorem. First Approximation Theorem (II): Let F(x) be a polynomial with integral coefficients which does not vanish for x=0 and has no multiple factors. Let p, a, X, in be real constants satisfying p > 0 , a>0, 0 < X < 1, O ^ j u t ^ l ; ! let ci, c 2, c8, c4 be positive constants; and let gf & 2 and gM ^ 2 be fixed integers. Finally let S = -fro1/ K\2\ K^,...} be an infinite sequence of distinct rational numbers K(k) = ?
* 0, where P(k)+ 0, Q(k)+ 0, (P(k),Q(k)) = 1,
with the following two properties. (A|j): For all k, t/d=l, ) ff
-
t/d=2,
132
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if d=3.
(B): For all k,
and Then for d=l,
v ^ \+p,
for d=2,
p+a ^ A. + JU
for d=3.
Proof: It suffices to apply the first form of the theorem to the sequence 2' and the zero or zeros $ , E obtained by Lemma 4.- By the same lemma, the new second form of the theorem implies also the original first form; both forms are thus equivalent.
Chapter 8 THE SECOND APPROXIMATION THEOREM 1. The two forms of the theorem.
This chapter contains a generalisation of the First Approximation Theorem which has just been proved. We begin by introducing some notations that will be used. If a is any real number, and /3 is any p-adic number, put I a I* = min(|a|, 1), so that always
|/3|* = min(|j8|p, 1),
Denote by P1,p2,...,pr; Pr+1»Pr+2»-">Pr+r»;pr+r'+l'pr+rf+2'"'' pr+r'+r" a fixed system of r+r'+r", =n say, distinct primes. It is not excluded that one, two, or all three of the numbers r, rf , and rlf , are equal to zero. Let further £*0, £i+0,..., £ r +0
denote a real algebraic number, a pi-adic algebraic number, etc., a pr-adic algebraic number, respectively. These algebraic numbers need not satisfy the same irreducible algebraic equation with rational coefficients, and thus they may belong to different finite extensions of the rational field. Next let F(x), Fi(x),..., Fr(x)
be r+1 polynomials with rational coefficients, which neither vanish at x=0 nor have multiple factors. It is not required that all these polynomials are distinct, that they are irreducible, or that they are non-constant. As in previous chapters, let again S ={/cv/, tf(2), «s3),...} be an infinite sequence of distinct rational numbers ,« = Z^
+0,
where P(k) +0, Q(k) *0, (P(k),Q(k)) - 1, H(k)=max(|P(k)|,|Q(k)|).
Finally, put pj
133
PJ
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and
r j=l
r+r* *
J
/ \
j=r+l
r+r'+r" 3
]=r+r f +l
3
and denote by Id, Ka, and r three positive constants. The Second Approximation Theorem can now be stated in two different, but equivalent forms, as follows. Second Approximation Theorem (I): If for all /r e s,
then T« 2. Second Approximation Theorem (n): If for all /r 'cS, *
Let S satisfy the hypothesis of Theorem (2,1), and let g, gi,..., gr be the corresponding real or pj-adic algebraic numbers. Denote by F(x), FI(X), ..., Fr(x) the irreducible monic polynomials which have g, gi,..., gr, respectively, as roots. We begin by showing that there exist positive constants y> ri>-..,rr such that, for all k,
and
(2):
(FjOc(k)) | J < yj U(k)- gj |*j
(] = 1,2,...,r).
Consider, for instance, the inequality (1). If E contains no infinite subsequence S such that (3):
lim |ic^k)-g|* = 0,
then the greatest lower bound M=
THE SECOND APPROXIMATION THEOREM
135
is positive, and it is obvious that for all k, |F(« ( k ) )l**rl« ( k ) -«l*
where
y-£»l.
Next assume that S does contain an infinite subsequence £' with the property (3). Then £ is a zero of F(x), and hence where G(x) denotes a certain monic polynomial with real coefficients. Denote by yo > 1 a number such that |G(x) | < y0 for all real x such that |x-g I < 1 and hence |x |< U 1+ 1. We have then, for every k. either
U^-4 I > 1
or
1
and hence, trivially,
|F(/c^) |%y 0 k^-{
and hence
|F(«(k))|<
I* = yo,
whence * min(l,yok (k) -S I) *y 0 k (k) -« I*, proving the inequality (1). Each of the inequalities (2) can be proved in exactly the same manner. From (1) and (2), it follows now that But, by hypothesis, so that, for all k, (k) T where Ka = y0y! ... y r K lB #(K^) « KKaH " , The assertion T< 2 of Theorem (2,1) is therefore a consequence of Theorem
3. The Theorem (2,1) implies the Theorem (2,11).
Let S satisfy the hypothesis of Theorem (2,11), We procede in a similar manner as in § 2; but it now becomes necessary to replace £ by a system of successive subsequences So, Si,...,Sr. If the lower bound L= inf
|F(K^)|
is positive, put and denote by £ an arbitrary real algebraic number distinct from zero. If,
136
.
LECTURES ON DIOPHANTINE APPROXIMATIONS
however, L=0, then S contains an Infinite subsequence S' for which |F(«(k)) | = 0.
lim
By Lemma 2 of the last chapter, there exist then an infinite subsequence So of Sf and hence of S, a real zero $ 4s 0 of F(x), and a constant yd ^ 1, such that U (k) -«ky 0 |F(« (k) )! forallK ( k ) eSo. It is thus obvious that, in both cases, there also exists a positive constant yo such that U (k) -d* such that
fc(k)) Ijj
a = 1,2,..., j-i),
for all K €S j-l- A further sequence Sj is now found by the following construction. If the lower bound Lj=
inf K
«Sj-l
is positive, put Sj = Sj_i and denote by £j an arbitrary pj-adic algebraic number distinct from zero. If, however, Lj=0, then Sj_i contains an infinite subsequence Sj such that
lim |Fj(K(k))|p. =0. J
Therefore, by Lemma 2f of the last chapter, there exist an infinite subsequence £j of 2] and hence also of Zj-1, further a pj-adic zero Ij of Fj(x), and a
THE SECOND APPROXIMATION THEOREM
137
positive constant r\ > such that U(k)- Ej lPj «yj lFj(*(k)) IPJ for all /c(k)eSj. In both cases it is therefore again obvious that there is a positive constant 7j such that k k U(k) - «, H * yj |F 3 ( (k) K ) |Jj
(k)k for all /c eSj.
By this construction, the elements of the final sequence Sr satisfy the inequalities
and hence also the inequality
But, by hypothesis, for all k,
and so, for all K 'eSr,
~, where K i Hence, on applying Theorem (2,1) to the sequence Sr, we obtain the assertion r^2 of Theorem (2,11). This concludes the proof that the two forms of the Second Approximation Theorem are equivalent. The analogous result for the two forms of the First Approximation Theorem was already proved in the last chapter. It will be shown in the next sections that also the First and the Second Approximation Theorems are equivalent. From what has been already obtained, it suffices to carry out this proof for the first forms of the two theorems. 4. The Theorem (2,1) implies the Theorem (1,1).
Let £, 5", p, or, X, JLI, g, gf, g", S, ci, c2, cs, 04 be defined as in Theorem (1,1), and let t) r ner+r! ff " - D e r+r f +l r+r+r Pr+1 -Pp+ri » g - p r+r f +l - p r+r f +r" be the factorisations of g, gf, and'g", respectively, into products of integral powers of distinct primes. If X<1 and jii < 1, and if we are dealing with the cases d=2 or d=3 of the theorem, we deduce from the hypothesis that and hence that all primes pi, P2,...,pn are distinct; here again
138
LECTURES ON DIOPHANTINE APPROXIMATIONS n = r+r f +r ff .
For, first, P^ and Q*^ are, for sufficiently large k, divisible by gf and g", respectively; hence (gf, g f t ) = 1 from (P*k\ Q^) = ll Secondly, it was already found in §1 of last chapter that for X < 1 necessarily (g, gf ) = 1. Third, if ji < 1, then we have. also (g, g") = 1. For otherwise g and g11 would have a common prime factor, pi say. Then the hypothesis would imply that lim |je W -«il p l =0, k —» °°
lim |Q (k) l pl =0, k — »°°
where $1 denotes the pi-adic component of S. However, these two limits contradict one another; for the pi-adic values IK ' |P1 are bounded by the first limit, but tend to infinity by the second limit. These remarks no longer hold when X = 1 or y, = 1, and they become unnecessary in the remaining case when d = 1 because then no g-adic number E occurs. We shall simply put
r = 0 if d = 1, and
r1 = 0 for X = 1 and r11 = 0 for p, = 1 if d = 2 or d = 3; this corresponds to disregarding trivial inequalities. The proof of Theorem (1,1) procedes now as follows. The g-adic, g'-adic, and g"-adic pseudo- valuations that occur in the statement of this theorem allow upper bounds in terms of products of pj-adic valuations. Thus, by definition, ' logg
logg
and hence ejlogpj
Therefore,
We find in just the same way that
(5): and
r;
n |P j=r+l
THE SECOND APPROXIMATION THEOREM
139
r+r!+r" /. v /. v H |Q (k) U«|Q (k) | g ". Pj j=r+r'+l -
(6): Here the products r
/. »
n |«w-«, I-., j=l
r+r1 /.(k)v
n
* j=r+l
|P lpjJ ,
r+r'+r"
/. »
n |Q<%J
j=r+r'+l
and the pseudo-valuations \K(k)-S\e, |P(k)|g'( (Q(k)|g» are interpreted as meaning 1 if r=0, r?=0, or r"=0, and hence g=l, g1-!, or g"=l, respectively. In the case d=2 the formulation of Theorem (1,1) does not involve any real algebraic number £ . We may then denote by £ an arbitrary real algebraic number not zero and may also use the trivial formula l*(k)-| |* * 1.
We finally note that, in the cases d=l and d=3.
and in the cases d=2 and d=3,
as soon as k is sufficiently large; for the expressions |jcW-{|, and U W -Cj|p
(j= 1,2,..., r),
respectively, tend to zero as k tends to infinity. It follows therefore, by (4), (5), (6), and the hypothesis of Theorem (1,1), that all but finitely many of the elements /c(k) of the infinite sequence £ satisfy the following inequalities: For d=l: where
For d=2: where For d=3: Where
*(K(k)) * clH(k)^.l.c,H(k)X-1.c4H(k)'i-1 =KlH(k>-T Ki = cic3C4, T = p-X-/i+2.
.Oc«) * iW^-CsH^-1. c4 KI =c2c3c4, r = a-\-/x+2. ®(K(k)) * c^-^E^^RKI= Cic2c3c4, T = p +a-X-/jt+2.
Here, in agreement with the earlier selection of rf , r11, gf, and gM, it is necessary to put cs=l if A.=l and C4=l if jLt=l. Theorem (2,1) states in all three cases that
T^ 2.
140
LECTURES ON DIOPHANTINE APPROXIMATIONS
Hence p ^ X + n for d=l; or < A. + jut for d=2; which is the assertion of Theorem (1,1).
p + a< X + JLI for d=3;
5. The integers ej.
In this section and the following ones it will be proved that, conversely, Theorem (2,1) follows from Theorem (1,1). This proof is a little more difficult. It is indirect and requires that the sequence be repeatedly replaced by a suitable infinite subsequence. /,x Let us assume that all elements K of the infinite sequence £ satisfy the inequality but that T > 2.
Hence r may be written as T =2+26
where € is a positive number; without loss of generality, 0<e
« H
- •
if j = r+l,r+2,...,r+rf, if j = r+r'+^r+r'+^.^r+r'+r".
Put
so that
THE SECOND APPROXIMATION THEOREM
141
(k) Since Hv ' tends to infinity with k, and since r~2+2e, it follows that
for all sufficiently large k. On replacing, if necessary, S by an infinite subsequence, it may be assumed that this inequality holds for all k. /. x By the hypothesis, all n primes PJ are distinct, and all numerators P and denominators Q are different from zero. It follows then from the basic inequality for the valuations of a rational integer that
and hence that r+r?
/b-\
t\*\ 1
r+r'+r"
1 n ipWlp^nW, . n! P
j=r+l
J
Thus, from the definition of aj, r+rT /.v ]=r+l
j=r+r +l
/.*
|Q(k>|
r+r f +r M /, v j=r+r ? +l
and hence
j=r+l Next put
Then, for all k,
and
because
s-2
is an increasing function of s, and 0<e^l. It is, in particular, (k) evident from these relations that all a j are bounded, s
Q ^ a^
*$ a;
here a is a certain positive constant that does not depend on k.
142
LECTURES ON DIOPHANTINE APPROXIMATIONS
Let N be the positive integer
so that 1 + £ log Pj < V 3=1
Further put
Since 0 « e(jp < [a N],
the system of the n+1 non-negative integers \6o
9
ei ,..., en /
has not more than possibilities. Hence there exist an infinite subsequence S' = {/c(il\ K(l2*; K^l3\...} of S, where u < i2 < is <..., and a system of n+1 non-negative integers \eo, ei,...,e^Jwhich are independent of k, such that eo = eo, ei K = ei,...,e = en for k=l, 2, 3,... . Since we may, if necessary, replace L by Sf, there is no loss of generality in assuming from now on that S1 is identical with S. This means that, for all k, fa} (k} ejw - [Ojw N] - ej (j=0,l,...,n), and hence ej « ajk)N < BJ + 1
(j = 0,1
n).
Thus, first, e + ? e lo • < faik) + ? a^k)lo V = N
THE SECOND APPROXIMATION THEOREM
143
whence n. -| N)< e0 + E e j lo SPj * N
1 + E logPj < ^j~-
because
Secondly, r
n
E
and therefore
+|
v
ajk)logp^N < (<*> + E ejlogpj + (l + £ logpX
whence
6. The numbers g, gf, g f f , p, o,K,n. Now put again r+1
_ -r+r'+l -
-r+r.»
where, as before, empty products mean 1. Here we may disregard prime factors PJ that belong to exponents ej equal to zero. The inequalities
/;(1 - g*w )N < e + v 2j j iSPj ^<M > e lo
N
0
j=l
f , ^*N
e
o + L e j lo gP] > "H" 3=1
just proved are equivalent to (7): (l -^)N < eo + logfeg' g f f ) « N,
e0 + logg > ^
> 0.
They imply that at least one of the two numbers p = —j^-2-,
(8):
o- =
1^
,
which evidently are non-negative, must be positive. By §5, we have fl
IK
\
)|e
—c |
/I \
•"• ii
(lf\
*•*/
/I \
=
(]r\ (lr\
/« V /OV
JjL
/
/lr^
R' 'p«
^ id
**
IN
^lr\
^ Jti
(2-4-eW
V*''^C/C0
N
This result may be strengthened to (9):
U (k) -£l ^H (k) " p
ifp>0.
/| '
=
H
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LECTURES ON DIOPHANTINE APPROXIMATIONS
For, by hypothesis, H (k) £ 2 and hence H(k)"p < 1. Similarly, again by §5, for ] = 1,2,...,r, _..W. (k H >
N
(2-K-)ejlogpj "- «„«
Assume, for the moment, that cr> 0, hence that ei,...,er do not all vanish. We may, simultaneously, renumber the primes pi,...,Pr and their exponents ei ,..., er. It may thus be assumed that, say, ej > 0 f or j = 1,2,..., u9 but 6j = 0 f or j = u+1, u+2,...,r. Here 1 < u < r , and g becomes now the product Denote by E the g-adic algebraic number S-— (Si,...,5u) with only u components; by the hypothesis, none of these components is zero. Just as in the proof of (9) we find that (2+€)ejlogpj ( (k k %lpj*H >~ N (J = 1,2,...,U). Since, by definition, logg
logg
it follows then that (2-K)lQgg W
CIO):
U -ff|g«H
W
N
=H-
f f
Next put (II).
VIA;.
x-1 lOgg* A - i - (2+6)j^.
,
lOgg" / X -- i - (2+6)jq.
It was shown in §5 that r+r1 /. x r+r'+r11 /. x E a^logp^l, E j=r+l J j=r+r'+l Therefore r+r' r+r' /.» /.» ' M r+r' /./. v logg'= E eJr log Pj *N E « Wpj=-|r E a3 J J J W j=r+l j=r+l s j=r+l
if0.
THE SECOND APPROXIMATION THEOREM
145
and similarly
so that (12):
O ^ X ^ l , O^jii^l.
In particular, the equation X = 1 holds exactly when gf = 1, and the equation /i= 1 when g"= 1. For j = r+1, r+2,...,r+r!, M M to (2+e)ejlogp1 |p(k)|p. = H(k)-ajk)logPj . H(k)-«jkVk)logPj * H(k)' £—X
m
Hence, from / logg' logg' \ |p(k)|g =max^|P(k)|p^l^Pr+l ,..., |p<%*r+rUogprH.r^ and from the definition of X, it follows that ipWlg^H*^ 1 .
(13):
For X = 1, hence g' = 1, the proof of this inequality no longer holds; but the inequality itself remains valid for any integer gf ^ 2 because the gf -adic value of a rational integer cannot exceed 1. An analogous proof leads to the inequality |Qw|g" ^
(14):
where, for JLI = 1, gM may again be replaced by any integer ^ 2. 7. The Theorem (1,1) implies the Theorem (2,1). The proof of the Theorem (2,1) can now easily be concluded. By the formulae (9), (10), (13), and (14) of § 6, the sequence S has the properties Ad and B of the last chapter; here
d = 1 if p > 0, a = 0;
d = 2 if p = 0, a > 0;
d = 3 if p > 0, a > 0.
Now it follows in all three cases, from (7), (8), and (11), that p+ff,X^=(2H-e)^+1°ygt>)-2> so that, by 0 < e < 1,
However, by Theorem (1,1),
146
LECTURES ON DIOPHANTTNE APPROXIMATIONS p<X
+ j u i f d = l;
cr <X + JLL if d = 2;
p + a < X + / j t i f d=3.
The hypothesis that r > 2 leads therefore to a contradiction and is false. This concludes the proof.
Chapter 9 APPLICATIONS 1. The theorems of Roth and Ridout.
The two Approximation Theorems proved in Chapters 7 and 8 contain as special cases the theorems of Roth and of Ridout. These results were already mentioned in the introduction to Part 2, and we then gave the references to the literature. Roth's theorem may be considered as either the special case d=l, X=l, /i=l of Theorem (1,1) or as the special case r=r f =r"=0 of Theorem (2,1). It states: If £ is a real algebraic number; if p and ci are positive constants; and if there exist infinitely many distinct simplified fractions T such that
pv
•c,iQ w r p ,
Q then p ^ 2.
This theorem is obvious if £ is rational; thus, e.g. the case when £ =0 p(k) may be excluded. Now —— tends to £ as k tends to infinity. Hence, by Q 5 + 0, the integers P , Q Wf and H^) have the same order of magnitude. It follows that, for all k,
where Ci is a further positive constant. Hence the assertion is an immediate consequence of either Approximation Theorem. We next show that Ridout's theorems may be deduced from Theorem (2,1). His first theorem is as follows. Let again £ * 0 be a real algebraic number; let p, ci, cs, and c4 be positive constants; and let X and ju, be constants such that
' "M " Assume that there exist infinitely many simplified fractions with the following properties:
p(k) —TT-V
»
(i): (ii): The numerators P ' and the denominators Q
147
are distinct from zero
148
LECTURES ON DIOPHANTINE
APPROXIMATIONS
and can be written in the form ]=1 J ]=rf+l J where pi,...,pr», Pr f +l>-.,Pr f +r tf are finitely many distinct primes, ei,...,er», er»+i,...,eri+ru are non-negative integers, and Px(k)*, Qv(k)* integers such that Tfeen For the three integers P , Q , and IT ' have again the same order of magnitude. It follows that there exist three positive constants ci, c'3, and ci such that, for all k, and
n |pW|p,J . c'aH^-1,
T'lQ^UJ
j=l J=r'+l The latter inequalities hold because, e.g. 3=1 Pj
PW*
Cs
"
'
C
*
Therefore, from the hypothesis, p(k) (k)
n |P
J
and so the assertion is contained in the special case r=0 of Theorem (2,1). Ridout's second theorem generalises that of Roth in a different direction. It is essentially equivalent to the case r f =r"=0 of Theorem (2,1), and it may be stated as follows. Let $, $i,,,,£ r be a real, a pi-otftc,...,a pT-adic algebraic number, respectively. Let T andc be positive constants. Assume there exist infinitely many distinct simplified fractions such that *H (P (k) -|]Q (k) lJj * Then r ^ 2. As we see, this theorem does not require that
M-T
APPLICATIONS
149
a restriction imposed by both Approximation Theorems. However, this presents no difficulty. For choose a rational integer n > 0 such that the r+1 algebraic numbers {' = £ + n,
|j[ = £1 + n,..., £r = %-p + n
are all distinct from zero, and put P< k >'=P ( k ) + nQ ( k >. Then H(k)' * H(k). (nn-1),
H(k>' = max( |P(k>'|, |Q« |),
where
and it is evident that T p(k)
_
f
*_
Q^
Q
and
PJ
. inw_| '
Further
and hence
because
The hypothesis implies therefore that
r
n where Ki denotes the constant & = c(n+l)T sup 5 iQ^lp 1 K
J=I
where the supremum exists because (P«Q (k) ) = land «;*0.
150
LECTURES ON DIOPHANTINE
APPROXIMATIONS
On applying now Theorem (2,1) to this inequality, with r T =r tT =0 and with 1*9 ££>•••>f £~ as the algebraic numbers, the assertion r * 2 follows at once. (k) r For P can be zero for only finitely many k. 2. The continued fraction of a real algebraic number.
One simple application of Ridout's first theorem is of interest in itself. Let | be a real irrational algebraic number. Then | can be written as an infinite continued fraction £ = [ao, ai, aa,...] where the incomplete denominators ao, ai, a^,... are integers, all but perhaps ao being positive. Denote by -^ the n-th convergent of this continued fraction. As was proved in Chapter 4,
We can now show the following result. Both the greatest prime factor of Pn and that of Qn tend to infinity with n. For assume, say, that there exists an infinite sequence of suffixes, N = {m, na, n3,...}, where n4 < ifc < ns < ..., such that, for neN, Pn allows only the finitely many prime factors pi,...,prf. We can then apply the first theorem of Ridout with p = 2, X = 0, p, = 1,
and evidently obtain a contradiction. The assertion for Qn is proved in the same manner. 3. The powers of a rational number.
The case X = ju = 0 of Ridout's first theorem implies the following result. Let £ 4 = 0 be a real algebraic number; let pi,...,Ps be finitely many distinct primes; and let e be an arbitrarily small positive constant. There exist at most finitely many systems of s integers, {e}={ei, ea,.«, es}, such that Ip^.pJ'-tKe-* where
APPLICATIONS
151
For put
**...<•-! where P and Q are positive integers which are relatively prime. To each system {e} there belongs then such a fraction - such that
II-*!<•-" It is obvious that
where p0 denotes the largest of the primes pi ,...,Ps- Hence I? - £ | < Q"p,
where
p
•*— ^ «
The prime factors of both P and Q are bounded. Therefore the first theorem of Hidout may be applied with p > 0, X = jit = 0, so that p > X + p,, p and it follows that the number of solutions £ , or, what is the same, that of systems {e}, is finite. This result may now be generalised, as follows. Theorem 1: Let % * 0 be a real algebraic number; let pi,...,p& be finitely many distinct primes; and let e be an arbitrarily small positive constant. There exist at most finitely many systems of s+1 integers {e}={e0,ei,..., eg} with e0 * 0 such that 0 < lpi e i ...p| s -e 0 «l<e" € E where E = . max |eJ . j=l,2,...,s J Proof: We assume that the assertion is false, hence that there are infinitely many different solutions fjf-' \6o
PV. ;•••> ' rf^' , 6l "g J 1
- 1 9 q ^ \Krir — l,o,O,.../
of the inequality (1): where
Jk) , E (k) e(k) w 0< Ipf1 ...p^s -ertKe" 6 " 1
152
LECTURES ON DIOPHANTINE APPROXIMATIONS
For each solution {e&)} put
e •
00
with positive integers P and Q which are relatively prime. We divide now the solutions into equivalence classes by putting two solutions {e } and {e T- into the same class, in symbols whenever p(k) =
JW 0(k) e0 Q
e0
/ \ Each such class evidently contains a unique element, {e } say, for which the positive integer |eo I is a minimum. This solution {e^mfy is said to be the minimum solution of its class. Consider now an arbitrary solution {e } for which the greatest common divisor
is greater than 1, and put
so that
(-,«*. p«Vi. Then W*
_<W* -#
efe)*
"
(k)* (k)* with new integers ei ,..., Cg . From the construction, it is clear that, for 1 = 1,2,..., s, ^ (k) • (k) ^ n (k)* (k) (k) . ft ^ (k)* 0 < e.7 < ej 7 if e. ^0, ej = e. 'if34!e: < 0n . Therefore iuttA
ir (k)*-
J-lX^-»B ' satisfies the inequality
APPLICATIONS
0«.
153
* (k) E
It follows then from (1) and (2) that e(k) e(k)* (k)* e-^^ l * 0 < ,|pf ... p!s - eik) % |, < 2-rr|dw| and hence {e
£EW* < e'eB ,
} is likewise a solution. Evidently
{.«V{.«J
and
|e*|<
k«|.
We have thus proved that every minimum solution {e^m'} satisfies the equation (e<m), P(m>) = 1,
(3):
Next consider any solution {e } which is in the same class as the minimum solution {e
}; i.e.,
Then, by (1), ,,=.&)
p(m) 0<
eTV517 Hence there can only be finitely many solutions {e } ~ {e } because otherwise the right-hand side would tend to zero. It follows that it may be assumed, without loss of generality, that all solutions {e } are, in fact, minimum solutions; for we may, if necessary, omit all solutions which are not. Next, we are allowed to renumber the primes pi,..., pg, and to replace the infinite sequence of minimum solutions {e } by any infinite subsequence. There is then no restriction of generality in imposing the following further assumption. There are two non-negative integers rf and r" with r f +r M > 0 such that, for every solution {e 7, ej k) ^0 if j = 1,2,..., r'; This implies that, for every k, P Pi>*->P r '> and Q Finally put
ejk) ^ 0 if j = r' + l,r'+2,...,r'+r".
(k)
cannot have prime factors distinct from
cannot have prime factors distinct from Pr' + i »...,p r T +r
154
LECTURES ON DIOPHANTINE APPROXIMATIONS
It Is clear that H*' tends to infinity with k, and that *
p(k)
e-eEfe>
0<
(4):
Q By the construction, P (3) that also
and Q
axe relatively prime; it follows then from
(P(k),Q«*) = 1.
(5):
By what has just been said about the prime factors of P
1=1 It is further-obvious that
and Q
,
J
r f +r"
o<
n
j=r' + l Therefore 0< |Q(k)
*J
j=r'+l whence
,(« (6): 0<
P
Q
p(k) Now £ + 0, and it follows from (4) that the fractions j£7gT* tend to g. Therefore P and Q have the same order of magnitude as H . There exists then a positive constant c such that, for all k, |p(k)Q(k)% cH^2. (k)
(lei*
M
On the other hand, the definitions of PW,QW , and Hw imply that
where again po denotes the largest of the primes pi,...,ps. Hence w ii(k)-p
where slogpo
APPLICATIONS
155
whence, by (6), 0< -f-* •* Q Here, for shortness,
On applying now Theorem (2,1) with r=0, we obtain a contradiction. This concludes the proof. By way of example, let £ £ 0 be a real algebraic number; let u and v be two integers such that u > v £ 2, (u,v) = 1, and let e > 0 be an arbitrarily small positive constant. From Theorem 1 it follows immediately that there can be at most finitely many pairs of integers {e0, ei} with e0 * 0, ei ^ 1 such that
The special case $ = 1 means that the fractional parts of the integral powers of - cannot be too small. This result is useful in the theory of Waring's problem. For it allows to prove, for all sufficiently large positive integers n, that every positive integer is the sum of not more than
n-th powers on ,311 uon , in i ,4 o ,...,
where certain positive integers do, in fact, require this number of n-th powers1. 4. The equation P+ o+R
= 0.
This section deals with a general theorem on triplets of integers. Theorem 2: Let c and v be two positive constants, and let pi,...,Pr;Pr+l'*"> Pr+r'» Pr+r1 +!>•••» Pr+r'+r1 be finitely many distinct primes. Denote by S an infinite sequence of distinct triplets {p(k),Q(kU(k)}
(k = 1,2,3,...)
where P^ , Q*', and IT ' are integers as follows, 1. See Hardy and Wright, Theory of Numbers (Oxford 1954, 3rd ed.)« 335-337; K. Mahler, Mathematika 4 (1957), 122-124.
156
LECTURES ON DIOPHANTINE APPROXIMATIONS P» + 0, Q(k) + 0, R(k) + 0, P« + Q« + R(k) = 0, )> Q(k)} = (p« R(k)} = (Q(k)> R(k)}
Put
and write P(k*, Q^\ and R^ as products of integers, P« = P^fP,
Q(k) - Q^OJk),
R(k) =
R^,
where Pi has no prime factors distinct from pT+i,...,pr+ri, Qi no prime factors distinct from Pr+r'+l'—'Pr+r^r1*' a^ Ri has no prime factors distinct from pi , ..., pr . # (k= 1,2,3,...,), z/ ^ 1. Proof: For each k, either
(7):
|P (k) IHQ (k) |* |BWI,
or one of the five inequalities obtained from (7) by permuting P , Q , R is satisfied. Since we may replace 2 by any infinite subsequence, and since we are allowed to rename these three letters and, at the same time, the corresponding sets of primes, there is no loss of generality in assuming that, in fact, the inequality (7) holds for all elements of S. Put now (k) P(k) w and $ = 5i = - = 5r = -1. and, just as in the last chapter, write
n j=r+l
j=r+r'+l
Then, by (7),
,<w
(k)
Further also
,(k) QM
* -- |RIfi^l Pj 'Pj
For either PJ is a divisor of Q(k*, and then it is prime to R^ and so
APPLICATIONS
157
(IT)
or PJ is prime to Q , and then
R(k)
_- |IB R (k)i* Ipj_- i-Wi IB |p] .
Therefore
n U W -{jlw- n lR (k) L., i—l
*
1—1
and it follows that r
n
/. v r+r? /.v r+r'+r" /. x lipw I n |P |T>W| TT lr*W||p.. IR Ip4. n |p.. n |Q
3 =1 j=r+l Q Next, the hypothesis implies that, e.g. r
,
(]r\ , -D^K'
n IK
r
,
(lr\ , K
,
3
r
^lr^
j=r+r'+l
Tlr^ K
*D> ' 13W ^ <: TT T3^ ' I |p. =- nIT t(\tii |p..|K2 Ipj^ n IRi Ij
and, in the same way, r+r1
n I
3(k)
r+r+r"
n
j=r+r f+l
]=r+l Therefore
I
R(k)
^« cH
Finally, by (7),
H(k) = |p«, = (Q (k) +R ( k) , ^ |Q(k), + |R(k),, 2 |Q(k), 9
so that
IQ I 55 2 H ' and hence
It follows then from Theorem (2,1) that whence the assertion.
3
(k)
158
LECTURES ON DIOPHANTINE APPROXIMATIONS
It is again not difficult to construct examples which show that the assertion z/^1 of the theorem is best-possible. The Theorem 2 is a special case of the more general Theorem 3: Let F(x,y) = Foxf + FiX^y + ... + Ffyf, where f ^ 1, F0 * 0, Ff * 0, be a binary form with integral coefficients which has no multiple factors. Let pi,..., pr, P r+ i,—, Pr+r»> Pr+r f +l>— 9 Pr+rf+rM be finitely many distinct primes, and let c and co be two positive constants. Finally let 2 be an infinite sequence of distinct pairs of integers {P(k), Q (k)jguch thatfforall k, P(k) * 0, Q(k> * 0, (P(k),Q(k)) = 1, H(k) = max ( | P(k) , |Q(k) |) and
j. n IP (k) liPr3
n
lo (k) l|D. <»
j=r+l * Then co ^f-2. The reader should have no difficulty in deducing Theorem 2 from the special case F(x,y) = x+y of Theorem 3, and in proving Theorem 3 by means of Theorem (2,n). 5. The approximation by rational integers.
The two Approximation Theorems proved in Chapters 7 and 8 have analogues relating to the approximation of integral g-adic numbers by rational integers. There are again two such theorems, each one having two equivalent forms. These theorems are as follows. Theorem (4,1): Let S-~-(£ i,..., |r), where |i + 0,..., £r + 0, be an algebraic g-adic integer. Let gf ^ 2 be a positive integer9 and let c2, c3, a, and X be four real constants such that c2 > 0, c3 > 0, a > 0, 0 ^ X ^ 1. If there exists an infinite sequence S = {pW, p(2), P^,...} of distinct positive integers such that, for all k,
then a < X. Theorem (4,n): Let F(x) be a polynomial with integral coefficients which does not vanish at x=0 and has no multiple factors. Let g s* 2 and g1 ^ 2 be two positive integers, and let c2 , c3 , a, and X be four real constants
APPLICATIONS
159
such that ci > 0, c8 > 0, 0, 0 ^ X < 1.
If Mere exists an infinite sequence S = {pW, P< 2 ), P^,...} of distinct positive integers such that, for all k,
then ff < X. Theorem (5,1): Let pi,..., pr, Pr+i,...,Pr+rf be finitely many distinct primes, and let gi * 0,..., £r * 0 be an algebraic pi-adic integer, etc., an algebraic pr-adic integer, respectively. Let Ki and r be two positive constants. If there exists an infinite sequence S = {P^\P(Z\P^3\...} of distinct positive integers such that, for all k,
3=1
j=r+l
then r * 1. Theorem (5,n): Let pi,...,pr, pr+l>...,Pr+rf be finitely many distinct primes, and let Fi (x),..., Fr(x) be r polynomials with integral coefficients which do not vanish at x=0 and have no multiple factors. Let KB and r be two positive constants. If there exists an infinite sequence S = {pW, pW, p(3),..-} of distinct positive integers such that, for all k, j=r+l then T ^ 1. A discussion similar to that in Chapters 7 and 8 allows again to show that these four theorems are equivalent, in the sense that each implies the other three. It suffices then to prove Theorem (4,1). This is done by essentially repeating those constructions and estimates of §§2-8 of Chapter 7 that led to the case d=2 of the Main Lemma. One assumes that the assertion is false and that, say, a = X + 4e where 0 < e ^ -= . The proof then precedes with the values
and hence with the values = TT » Qh = 1, Hh = Ph
(h = 1,2,..., m).
Here the parameters m, s, t, ri,..., rm are selected just as in §2 of Chapter
160
LECTURES ON DIOPHANTINE APPROXIMATIONS
7, and the polynomial A(xi,...,xm) is defined as in §3. In particular, the inequality (14) of §2 takes the form,
The proof now depends on upper and lower estimates for the integer A
(l)=l fl)=A lx...l m ( ' Cl '-'' Cm) * 0 ' which has the explicit value
a<
i
mmmi
(MPI1'11 Pmim-lm "" V * •"
Here, by the upper bound for ailmmm im,
* 5(4c)mri (2ri+1-l)...(2rm+1-l) < 5(4c)mri (22ri)m = 5(16c)mri Hence it follows that, in the notation of Chapter 7, m
lA/,,1 « (80c)mri max &**... P^'lm * (SOc)^1 max (Del (i)el and therefore
The lower estimate for the case d=2 of N(j), obtained in §7 of Chapter 7, still remains valid; but since N(i)=A(i), it takes the form
On combining these two inequalities, we find that P!E < (80ccia)m, where we have put E = (l-A+orJSi-d+ejSa. In explicit form, E = (l-X-wr){|(m-s)-A} - (l+€)(|(m+s)-A} = = (cr-X-e)m -
(or-X+e+2)s - (or-X-e)A
APPLICATIONS
161
Further, just as in Chapter 7,
_ em
o
6m
It follows then that E = -5.3e.m - -g(5e+2)s - 3e-A £ -sr- 015" 5 If" ^
em
•
Hence, if Pi was chosen so large that
1 6
Pi £ (SOcda) , a contradiction arises, proving the assertion. 6. An example. By way of example, let p=pi be an odd prime, and let a denote one of the p-2 integers 2, 3,..., p-1. Put p (k) = a pk-l
(k = 1,2,3,...) .
It follows from Euler's theorem that
and hence P(k+1Wk)(modpk), or, what is the same, /o\
(8): In particular,
|-n(fc+l) |P - n(k)| P Ip ^^ P~-k
p(k) s p(k-l) s < > > s p (») s pW = a ( m o d p ) and therefore (9)
|P(k)-a|"P 2
p
8
By (8), the sequence {P^, P^ ', P^ ',...} is a p-adic fundamental sequence;
let a= lim P(k)(p) be its limit. Since Pvim'= PX"/F, evidently
/lim a = limP (k+1) =[ lim \kr>«
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LECTURES ON DIOPHANTINE APPROXIMATIONS
so that a is a root of the equation xP-x = x(x-l)(xp"2+xP"3+...+ X+l) = 0. Now a is distinct from 0 and 1, and by (9)
Hence a(a-l)*0, and so a satisfies the equation
Thus a is an algebraic p-adic integer distinct from zero. It has the following further property. If e is an arbitrarily small positive number, then there are at most finitely many positive integers k such that tir\\. UO;:
UP^ <• 0a~6pk |a -a^ i|p <
For put r=l, $1= a, and denote by Pr+lwPr+r! factors of a. It follows that
a11 tne
distinct prime
j=r+l Therefore, by (10), |p(k)-«|p T |P(k)|pj ^ P(k)-T, where r = 1« > 1, j=r+l and so the assertion is an immediate consequence of Theorem (5,1).
Appendix A ANOTHER PROOF OF A LEMMA BY SCHNEIDER The lemma by Schneider proved in §3 of Chapter 6 may also be obtained by means of a different method. This method has the advantage of leading to a slightly stronger result. It is due to my former colleague, G.E.H. Reuter, now professor of mathematics at the University of Durham. 1. A special case of Taylor's formula with Lagrange's error term states that if f(x) is four times differentiate in a neighbourhood of x=0, then
f(x) = f(o) + f (o)y, +f n (o)5 r +f f!l (o)^ r JL.
A.
t*.
where £ is a number between 0 and x. Let us apply this formula to the function f(x)=log cosh x for non-negative values of x. Then f (x) = tanh x, f" (x) = cosh-2x, fm (x) = -2 sinh x coshr8x and
fiv(x) = 4 cosh"2x - 6 cosh"4x. The fourth derivative assumes its maximum when cosh x=/3~, and so flv(x) < o\
for all x^O.
It follows therefore that
and hence that (1):
cosh x ^ exp^x2 + -~x4)
if x >0 .
2. Let again ri,...,r m be m positive integers; let further s, pi,...,pm be m+1 positive numbers. We denote by N the number of sets of m integers (ii,...,im) satisfying the inequalities
(2):
0 * u« n,...p 0 * im * rm, £ j£- *(f-s) E JJ , h=1 p h
\^ / h = 1 Ph
or, what is. the same, the number of such sets satisfying
(3):
0 « U « n ..... 0 * im « rm,
That both systems (2) and (3) have the same number of integral solutions is obvious because the transformation (ii >•••> im) -* (ri-ii ,..., rm-im) interchanges their solutions. 163
164
LECTURES ON DIOPHANTINE APPROXIMATIONS
3. Denote by u a positive variable, and put
and
.ir f=0 m
Evidently, m F(u) = n Fh(u) .
In the sum for Fj1(u) replace i by r^-i and note that
rh-i _ " " rh =_ /!_" "_ rh \ Ph It follows that
1)
max cosh/u( — - ^-2 )) i=0,l,...,rh \\Ph Ph//
.
Now cosh x is decreasing for x ^ Q and increasing for x ^ 0. The maximum is thus attained both when i=0 and when i=rh, and hence Fh(u) « (rh+1) cosh f*£ . Therefore, by (1), m h=l
< (ri+l)...(rm+l) exp that is, (4):
F(u) ^ (n+l)...(rm+l) exp jj- ^ f^\1 L h=l V^* /
+^g E h=l
ANOTHER PROOF OF A LEMMA BY SCHNEIDER
165
4. By definition, the inequalities (3) have N integral solutions (U,..., im)« These inequalities may also be written as
and they therefore imply that
On the other hand, all terms in the multiple sum for F(u) are positive. It follows then from (5) that
F(u) £ N exp [ h=l On combining this inequality with (4), we find that
To simplify this estimate, put - m 1 m and fix u in terms of s by ca
The inequality (6) then takes the form (7):
N < (ri+l)...(rm+l) exp j-m (M B» - | ^t S4 For the applications it suffices to consider values of s with 0 < s ^ -n
and hence
s4 < 7 s2 .
It follows in this case from (7) that (8): N < (n+l)...(rm+l) exp (-Cms2) where C denotes the expression C - 2c* c* °* ~ c2 " 9c| ' 5. We finally impose on rh and ph the additional conditions
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LECTURES ON DIOPHANTINE APPROXIMATIONS
These inequalities evidently imply that
4--a- GET—©'• a)
4
and hence that
/_9\»
my
2 . 81 UP/ < c? . UP/ 121 ., 3 3^ 121 "TllN5 " ^ T ^ T W = ~8T ^ 2' UO/ UP/
°l«m//121\°
9
It follows therefore that
Thus the following result has been proved. Theorem 1: Let ri,...,rm be m positive integers, and let s, 6e m+1 positive numbers such that
are at most (rn-l)...(rm+l)e"ms2 integral solutions (ii,...,im) of the inequalities
0
* ii * ri.....°* »» * r- ,55 * S -B) h|
or, ^o^ is the same, of the inequalities 0 * U « *..... O . l m * rmi ^ JJ Let us compare this estimate with that given by the Lemma 2 of Chapter 6 in the special case when Pi = ri,...,pm = rm ! The notation is slightly distinct at the two places. If we return to that of Lemma 2, then, by this lemma, the inequalities 0 < ii < ri,..., 0 < im < rm, t rh ^ « 2|(m-s) (or ^2^ h=l have not more than
ANOTHER PROOF OP A LEMMA BY SCHNEIDER
167
integral solutions, and by Theorem 1 not more than
It is easily verified that always
r*\2
^e Wm / <^ V2m
~s~~ ' Hence Theorem 1 is not only more general, but also a little stronger than Lemma 2. Unfortunately, this improvement does not seem to be of great use in Roth's theory. 6. In Chapter 6 the Lemma 2 enabled us to prove the existence of the approximation polynomial A(xi,...,xm) which played such an important role in the proof of Roth's Theorem and the more general Approximation Theorems. Theorem 1 allows to construct a more general approximation polynomial. There is no need for giving the details of the proof which is just like that in Chapter 6. The final result is as follows. Theorem 2: Let F(x) = Foxf + Fix*"1 + ... + Ff ,
where
f £ 1, F0 * 0, Ff * 0,
be a polynomial with integral coefficients which has no multiple factors and does not vanish at x=0. There exists a positive constant c depending only on F(x) as follows. Let m be a positive integer, s a real number such that ms 2 ^log(4f);
O^s «|,
let ri,...,rm be m positive integers; and let pi^-^pm*^—j^m* Ti,...,T m be 3m positive numbers satisfying 10'
3'K
Then there exists a polynomial ri rm A(xi,...,xm) = % ... % a- i ii=0 im=0 '" with the following properties. (i): The coefficients are integers satisfying
and each coefficient
h=1
Ph
a^
h=1
<>ti
vanishes unless both
h=l
h=l
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LECTURES ON DIOPHANTINE APPROXIMATIONS
(ii): Aj lsi j m (x,...,x)te divisible by F(x) for all suffixes that
ji,...,jm such
0 « J i « n , . . , , 0 « j m « r m , £ Tf
h=l h
(ill): T/^e following majorants hold, A. . (Xl,...,xm)« cri+-+rm(i+Xl)ri...(l+Xm)rm, ji •••Jm
Should it be possible to replace Roth's Lemma in Chapter 5 by a stronger result, then Theorem 2 might become of importance.
Appendix B A THEOREM BY M. CUGIANI Roth's Theorem suggests the following problem. Let £ be a real algebraic number. To find a function e (Q) > 0 of the integral variable Q, with the property lim e(Q) = 0, Q— »oo
such that there are at most finitely many distinct rational numbers ry with positive denominator for which
Unfortunately, the method of Roth does not seem strong enough for solving this problem and finding such a function e(Q). A weaker result may, however, be obtained and was, in fact, recently found by Marco Cugiani1 . It states: Theorem of Cugiani: Let % be a real algebraic number of degree f; let e(Q) = 9f(logloglogQ)'T;
pW p(0 pO»)
^^oft'o^'of5)'"*'
where
e
QM
Q
/,*
Q
/,v
< •"'
beanittf
-
finite sequence of reduced rational numbers satisfying
P(k)
(k)-2-e(Q-)
Then (k+1)
limsup logQ This theorem is thus an improvement of that by Th. Schneider2 which was mentioned already in the Introduction to Part 2. In this appendix we shall sketch a proof of the following theorem which contains Cugiani 's result as the special case X = JLI = 1. Theorem 1: Denote by £ * 0 a real algebraic number of degree f; by g? ^ 2 and gfl £ 2 two integers that are relatively prime; by X and n two real numbers satisfying 1. Collectanea Mathematica, N. 169, Milano 1958. 2. J. reine angew. Math. 175 (1936), 182-192. 169
170
LECTURES ON DIOPHANTINE APPROXIMATIONS 0 «£ X ^ 1, 0 ^ ]Lt< 1, X + J L I > 0; by Ci, ca, and cs J&ree positive constants; by e (H) $&e Junction e(H)= 5Vlog(4f) (logloglog H)~ *" ; and by S = {/cH fcH K^,...} an infinite sequence of distinct rational numbers K fe) = 4!
where P ee,
&e properties
|P(k)|g'
(2): Then
logH(k+1) lim sup Tj-v— = oo . k-*°° logHW 1. The proof of Theorem 1 is indirect. It will be assumed that S has the properties (1) and (2), but that the assertion is false; i.e., there exists a constant c4 > 1 such that, for all k,
Hence if X is any sufficiently large positive number, there is an element K ' of 2 for which H^H^^H04. From now on we put for shortness and denote by m a very large positive integer. We further put w 9 m-l m 2
t = e" -
,
| m8 X = et
and note that e (H) is given by e(H) = 5a(logloglogHr F . 2. By hypothesis S contains infinitely many distinct elements K , so that lim H^ = oo . k-*oo
A THEOREM BY M. CUGIANI
171
It is therefore possible to select m elements
(h=1 2
' .....m)
of S, of heights Hh = H(ih) = max(|Phl,lQhl)> ee, such that Xh^Hh^Xjj4
(h=l,2,...,m),
where
2
20^
2
20^
2
2c^
Xi - X, X2= Hif ^ XX * , X8= H/ ^ X2t ,..., Xm = Hm.i
It foUows that
whence, in particular, Hr< H2< ...
Further, for all h,
These inequalities, however, imply that (3): because
Hh^e __
e
em
(h = 1,2,..., m),
m-l\m m 9 m-l 2 e <ee
as soon as m is sufficiently large. From (3),
vm Hence the sum m 0-= E €(Hh) h=l
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LECTURES ON DIOPHANTINE APPROXIMATIONS
satisfies the inequality, a £ 5aVm . 3. Just as in §2, Chapter 7, define m-1 positive integers r2,...,rm in terms of a further positive integer ri by the formulae (rh-l)logHh < rilogHi < rnlogHh (h = 2,3,..., m). Here ri will be chosen so large that the quantity fcJ3!?L,m riirl is already so small that 0 <e <£—•< 1. Evidently rhlogHh= fl+^^j(r n -l)logHh< (l+0)rilogHi < 2rilogHi, hence 2r n _ilogH ft .i & 2rilogHi > and therefore
In particular, we find again that m ri> ra> ... >r m , £ rh* h=l 4. Apply now Theorem 2 of the Appendix A, with F(x) a minimum polynomial for f . The choice in §1,
,
ms 2 =a 2 =log(4f)
is allowed because m may be assumed so large that the additional condition of the theorem, 0* 8*5
is likewise satisfied. Next fix the parameters ph, ah> and rji of the Theorem by Ph = ffh = *h> Since
rh =^Ze(v^
(h = 1,2,..., m).
A THEOREM BY M. CUGLANI
173
the further condition of the theorem,
also holds provided m and hence X, Hi,...,Hm are sufficiently large. There follows then from the theorem the existence of a positive constant c depending only on £ , and that of a polynomial *i rm . A(xi,...,xm) = 2 ... 2 aixl ^xi^.x^O ii=0 im=0 "' m with the following properties, (i): The coefficients &iltu9i are integers such that
and they vanish unless
(ii): AJJ ...jm(£ v> 5) vanishes for all suffixes ji ,..., ]m such that 0«Ji«nf...fO«Jm
J JjL^(i. B ) E fj h=l Th v^ 7 h=l Th
(iii): The following majorants hold,
We next apply Roth's Lemma of Chapter 5 to the derivatives of A(XI ,...,xm) at xi= /ci,...,xm=/<m. This lemma is applicable provided that lm(m-l)(2m A^t ^m2 + l) C mri Hi ^ 2 and c ^ Hi , i.e. Hi ^ c For large m both conditions are satisfied because 2 s
m H i ^ X = eTl It follows then from Roth's Lemma that there exist suffixes li ,..., lm satisfying m . ^• m+1
0 « h* plf..., 0*lm*rm> £ £h « 2 h=l
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LECTURES ON DIOPHANTINE APPROXIMATIONS
for which the rational number Aft)
=
A^ ...lm(Ki >."vKm) = A^ um j
fe distinct from zero. Put again m
h=l r h " The choice of t implies now that
as soon as m is sufficiently large. 5. From here on the proof runs very similar to that of the case d=l of the First Approximation Theorem in Chapter 7. The slight change in notation with respect to s (which corresponds to — in the former proof) does not affect the discussion. Denote by CB, c6, and c7 three further positive constants that depend on £ , but not on m. Further let J* be the set of all systems of m integers (Ji,-Jm) such that m . /* \ m li** ji^ 1*1,...,lm ^ Jm ^ rm> 2j T u ^ l ^ " 8 ) u T. • h=l T h ^ /h=l Th Then, just as in §4 of Chapter 7, Aft) * E Aj^.j^ ,...,«)\W"AW (j)eJ*
(Ki-tf^.-dtm-^™'1™,
and here TI
rm
Z... E l ji=0 i =0 m
Now the Th were chosen such that .» " 1+ ^
(h = M,...,m) .
It follows then from the construction of Hh and rn that m a x ^ ^ . . . ^ ^ " (j)eJ*
1 1 1
^
1
m
max (j)eJ* h=l
(this inequality is continued on the following page)
A THEOREM BY M. CUGIANI
175
m
Here
m
«•! (>*«!)--*
g X+M '
and ^h^rh>|rh, 10 *
hence
E ^^2 £ h=l T fc h=l
Therefore
and so, finally, (4):
|A(1)I * (clC.)mri
(X ri ^ {S-8) Hr
(m+^) - 2A} .
6. We next express again
as the quotient of two integers N(i)+0 and D(i)*0 that are relatively prime. The discussion in §§6-7 of Chapter 7, specialised for the case d=l, may be repeated without any essential change and leads to the inequalities c6mri
and
On dividing these, it follows that
(5):
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LECTURES ON DIOPHANTINE APPROXIMATIONS
where E* denotes the expression E* = (l7. We finally combine the upper bound (4) for | AQJ I with the lower bound (5). Then we obtain the inequality HiE
(6):
where the exponent
after a trivial simplification, may be written as E=
Or8)* " t^MJmB - ^
em -
Now .
—,, - 7 =-,- , or £ 5aV~m~,, 0 < 0 ^ — Vm m
.
and hence
as soon as m is sufficiently large. Therefore (6) implies that ^>TE Hi ^ (CiC B C6C 7 )
,
contrary to the assumption that
-X
Hi ^X = e r when m is sufficiently large. This proves the assertion. 8. It would not be difficult to extend Theorem 1 to the more general case treated in the First Approximation Theorem. There may even be a corresponding analogue of the Second Approximation Theorem; but a proof of such an analogue would perhaps require new ideas. At present it does not seem possible to replace the function e(H) by any much smaller function of H. Such an improvement would require a stronger result on the zeros of polynomials in many variables than Roth's Lemma. 9. Two simple deductions from Theorem 1 have some interest in themselves and may therefore be mentioned here. Theorem 2: Let p be a prime and q an integer such that p > q £ 2, hence (p,q) = 1.
A THEOREM BY M. CUGIANI
177
Let N = {n\l\ ir% n^,...} be a strictly increasing sequence of positive integers such that n
,/neN,
where gn is the integer nearest to (~\ . Then n
lim sup
(k+D
Proof: For every positive integer n put
"»-£• *-S-* where dn - (Pn, gn«in) " Both dn and Pn are powers of p; Qn is divisible by qn so that n^logQn logq
, '
and it IB obvious from
1(5)°- -
that
2
lim |^ = 1.
It follows that there are three positive constants yi, y2, and y3 such that 0 < Qn < YI Pn < yi pn
and hence
n
and 1^1 -n ^u-1 ^ - 1 ^-u IQnlq * q * V2Qn > ° < Sn ^ VsQn
where
,^x loe/2\ u.^!iflZ
^
logp
'
i-u-JSU ** logp '
Here the upper bound for g^ is a consequence of the asymptotic relation
The lower bound for n in terms of Qn implies that for all sufficiently large n,
178
LECTURES ON DIOPHANTINE APPROXIMATIONS IQnlogp ^ . 91ogQn Vloglog n VlogloglogQn ' From now on let neN. By the hypothesis,
la. Qn
^ —exp (- 10nl°gP^ ^ Sn \ Vloglogn/
Kn W,
^ y Q-/*-9*0*0*0^^" 2^ We apply now Theorem 1, with
Since 5Vlog(4f) = SVlogT" < 9, the theorem gives
and from this, by logQn-logyi ( ^ logQn logp "* logq the assertion follows at once. 10. As a second application we construct a class of trancendental numbers which, in general, are not Liouville numbers. Theorem 3: 'Let g ^ 2 be a fixed integer, 6 a constant such that 0 < 8 < 1, {con} a» increasing infinite sequence of positive numbers tending to infinity, {j/n} a strictly increasing infinite sequence of positive integers satisfying (n = 1,2,3,...), and {an} an infinite sequence of positive integers prime to g such that
number
n=l
transcendental.
A THEOREM BY M. CUGIANI
179
Proof: Put
n=l
n=N+l
so that
The integers PN and QN are relatively prime because PN = aN +
2 an g1*"1* B &N (mod g) n=l
is prime to g. From the hypothesis,
and
Let now N be sufficiently large. Since o>n increases to infinity with n, it is obvious that (1-0) con Vloglogi/n
is an increasing function of n for n > N. Therefore oo
/
oo
(1-0) 0>n \
_ -nil*" .... 1*11 ^ E g^(1+VBgBgi57 E n=N n=N Z n=N
Further n=N
n=N
1 g
"
because the integers vn are strictly Increasing with n. Hence
180
LECTURES ON DIOPHANTINE APPROXIMATIONS
A
^ 1
(1-0) OJN logQN logg
for all sufficiently large N. Assume now that the assertion is false and that £ is algebraic, say of degree f . Then Theorem 1 may be applied with
X = 1, /i = 0, g" = g, ci = 1, ca = 1, while g1 is an arbitrary integer prime to g. But for large N, 5Vlog(4f) < |(l-0)a> N because CON tends to infinity. Hence it follows from the theorem that
or, what is the same,
lim sup N->~ There exist then arbitrarily large N for which
For these N,
n=N
n=N
and hence 0
g'<
:
n=N
But
n=N
n=N
whence 0
< RN <
g
-3 .(!_e) = const. QN .
However, this inequality contradicts Roth's Theorem, and we obtain the assertion.
Appendix C APPROXIMATION THEOREMS OVER ALGEBRAIC NUMBER FIELDS 1. Particularly the second part of these lectures was concerned mainly with the approximation of (real, p-adic, g-adic, and g*-adic) numbers by rational numbers. In this appendix certain results on the approximation of algebraic numbers by the elements of a fixed algebraic number field, of finite degree over the rational field, will be discussed. Without proofs, we shall state generalisations of the former Approximation Theorems. Detailed proofs would be rather long and involved; but there is no reason to forsee any essential difficulties. One important special case was already investigated by W. J. LeVeque (Topics in Number Theory, Reading, Mass., 1956, vol. 2, 121-160). He proved the following generalisation of Roth's theorem. Let K be an algebraic number field of finite degree over the rational field T; let ^ be a real or complex algebraic number not in K; and let T >2. There are at most finitely many elements K of K satisfying |{-jc| < H(K)~ T . Here H(K) denotes the height of K, i.e. the maximum of the absolute values of the coefficients of the irreducible primitive equation for K with rational integral coefficients. LeVeque's proof shows quite clearly which changes, as compared with Roth's proof in the rational case, have to made for the theory over K. The reader should also consult the paper by C. J. Parry (Acta ma the matica 83, 1950, 1-100). This paper established p-adic generalisations of Siegel's theorem on the approximation of algebraic numbers by other algebraic numbers. Due to the new method by Roth, many of Parry's results can now be improved. Unfortunately, there still are difficulties if one wishes to study the approximation of a given algebraic number by all algebraic numbers of a bounded degree that do not necessarily lie in a fixed algebraic number field of finite degree. 2. As was already mentioned in Chapter 1, the rational field F has, except for equivalence, only the following valuations, wo (a), | a I, |a|p; here p runs over all different primes 2, 3, 5,... . These valuations are connected by the basic identity wo(a) = l a i n |aLF . P Further every pseudo-valuation of r is equivalent either to a valuation of r, or to a pseudo-valuation of one of the two special types 181
182
LECTURES ON DIOPHANTINE APPROXIMATIONS
/ I 11 I IT\ /I I I 1-1 I r IP\ maxdal ,..., |al ), max(|a|, |a| *,..., |a| ) . Pi Pr Pi Pr Here pi,...,Pr are finitely many distinct primes, and Xi,...,X r are positive constants that do not affect the equivalence class. Important particular cases are the g-adic and g*-adic pseudo-valuations |a|g and |a|g* . Very similar results hold for any algebraic number field K, say of degree n over the rational field F. Let
K«, K('),..., K(n) be the conjugate fields of K, all thought of as imbedded in the complex number field. For any a in K denote by aft) its conjugate in K(0. As is usual, let the notation be such that the first ri conjugate fields
K(ri) are real, while the remaining 2r2 conjugate fields are arranged in pairs K« K« ...,
~(ri+i) K , ^K(ri+r2+i) of complex conjugate fields. Put / \2
/. x 9 ii = -I,z,...,r 2;
t fi « i « r l f if
so that ri+2r 2 =
g. =n. i=l
x
3. First, the Archimedean valuation |a| of T has exactly ri+r 2 distinct continuations on the field K, and these are likewise Archimedean valuations. Let K^ be the completion of K with respect to |a d; then K± is the real or the complex number field according as 1 ^ i « PI or ri + 1 ^ i ^ ri +r2 , respectively; and K{ has the degree gi over the real field P, the completion of T with respect to | a |. If N(a) denotes the norm over r of the element a of K, the different continuations \ot Ii of |a| are connected with N(a) by the identity, (1):
|N(«)| =
^ r2 |
Secondly, let p be any prime, and let |a|p be the corresponding p-adic valuation of T. The principal ideal (p) in K need in general not be a prime ideal. Let
APPROXIMATION THEOREMS
183
be its decomposition into a product of prime ideal powers. Here 7r(p) denotes the number of distinct prime ideals Pi,...,Pp(p) that divide (p), and the exponents e(|>j) are positive integers. In the usual notation, e(pp is the order of J»j, and the positiye integer f (J>j) defined by
is the degree of fjj here N(t*j) denotes the ideal norm of t>j. We put so that
1=1 There are now ff(p) distinct continuations of |a|p on K which we denote
like | a|p, they are all Non- Archimedean. These valuations may be defined as follows. If a = 0, put Next let a 4 0. There exists then a unique rational integer hj which may be positive, negative, or zero, such that the fractional ideal
is the quotient -
of two integral ideals m and n both of which are relatively
prime to fi> In terms of this integer, put -h
The different ftj-adic valuations satisfy the identity (2):
7r(p) . v lN(a)|p= ^ |a |{jfl' . Denote by Kp. the completion of K with respect to I a (j^; then K^ is
an algebraic extension, of the exact degree g(?j), of the p-adic field Pp, the completion of T with respect to |a|p. The field K$, is called the fjfield, and its elements are called pj-adic numbers. These Jij-adic numbers may be developed in series similar to those for the p-adic numbers, and they have similar properties as the p-adic numbers. Third, the field K has the trivial valuation Wo(a), and this is the only possible continuation of Wo(a), the trivial valuation of the rational field r.
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LECTURES ON DIOPHANTINE APPROXIMATIONS
4. From the basic identity for r, and from the product formulae (1) and (2), we find at once that the elements a of K satisfy the basic identity for K, II 2 | a | l? i . n |a P(|l) = wo(a) . • i=l V
(3):
Here p runs over all prime ideals of K, and
g(n) = e(p)f (p) is again the product of the order and the degree of ft. This identity implies the basic inequality for K (4):
ri+r2 r / x n |a If1, n |a | j:*J; 5* 1
if a * 0 is any integer in K.
Here pi ,...,pr are finitely many prime ideals of K, all distinct. This inequality is of the same importance for the approximation theory over K as was the inequality
r la I n la |p. ^ 1
if a * 0 is a rational integer
for this theory over the rational field. 5. Similarly as for r, any finite set of distinct valuations
of K is independent, and every pseudo-valuation of K is equivalent to one of the special form maxdal^,..., l « l i , ( Here 0« q ^ ri+r 2 ; i!,...,iq are distinct suffixes 1, 2,...,r1+r2; Pi,.">Pr are distinct prime ideals of K; and A.i,...,Ar are positive constants. Actually, the values of the latter do not affect the equivalence class of the pseudo-valuation. One, but not both, of the numbers q and r may vanish, and if then the other number equals 1, the pseudo- valuation becomes equivalent to one of the valuations that were discussed already. 5. Of particular interest are certain Non- Archimedean pseudo-valuations of K which are analogous to the g-adic value lalg of r and may be defined similarly. For let 0 be any integral ideal of K that is distinct from both the zero ideal (0) and the unit ideal n=(l). Denote by
its factorisation into a product of powers of distinct prime ideals PI ,,,., pr, with exponents ei,...,er which are positive integers. Let further e(fij) be the order of PJ, and let pj be the rational prime that is divisible by PJ.
APPROXIMATION THEOREMS
185
Naturally the r rational primes pi,...,Pr need not all be distinct. The 0-adic pseudo-valuation \a |g of K is now defined by the formula e(pi)logN(0) e(y r )logN(0) 6llogpl In 10 - max(|a (^ ..... |a Irf*1*** ), where N(0) denotes the ideal norm of 9 . In the special case of the rational field r, |a|g becomes again the g-adic value |a|g because Pi = (PI ),..., pr = (pr) now are principal prime ideals, and all the orders e(pi),...,e(fir) are unity. There exist elements y* 0 of K which generate principle ideals of the special form
where m and n are integral ideals that are prime to 0. Therefore
and hence e(yi)logN(8)
e(p r )logN(0)
Irl, - M It follows then that, similarly as the g-adic value |a|g, the 0-adic value has the following two properties, (I): |an|0 = (| a |g)n for all aeK and all positive integers n; (II): |aynlu = |a Ij(lrl0) n for aU «€K and all rational integers n. In the special case when 0 = pe is the power of a single prime ideal, e(y)ef(y)logp
H-iai,
elogp
,
and so |a|g is now equivalent to the 9- adic valuation |of||i. 6. The completion of K with respect to \a |g, Kg say, is called the 0-adic ring, and its elements are called 0-adic numbers. Let similarly Kfi,...,K$r be the completions of K with respect to \a Ipi,..., \ot \?r, so that they are the pi-adic,..., pr-adic fields, respectively, which we have already considered. Just as in the g-adic case, there is a one-to-one correspondence A-«+(oLi,...,ar) between the elements A of K$ and the sets (oti9...9ar) of one element ar in Kp , respectively. This correspondence is again preserved under addition, subtraction, and multiplication; and whenever division is possible, it is also preserved under division.
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LECTURES ON DIOPHANTINE APPROXIMATIONS
7. In order to formulate the assertions in a convenient form, the following notation will be used. First, we put |a I* = min(|a |, 1), |a |* = min( |a It, 1) (i = 1,2,..., n +ra), lalr, 1),
|a|0, 1). Secondly, let q be any integer satisfying Then denote by ii,...,iq any system of q distinct suffixes 1, 2,...,ri+r2 arranged such that 1 * u < i2 < ... < iq * ri+r 2 ; for q=0 the system is empty. Third, «, 0f, 3" are three integral ideals distinct from (0) and o = (l) which are relatively prime in pairs. Fourth, r, rf , r11 are three non-negative integers, and are r+r'+r" distinct prime ideals of K. It is not excluded that one or more of r, r1, rlf are zero. Fifth, Pu,..., Pi >
,
111}' < c" k|
APPROXIMATION THEOREMS
187
•Then
L Assertion (1,H): Let F(x) be a polynomial with coefficients in K which does not vanish for x=0 and has no multiple factors. Assume there exists an infinite sequence of distinct elements K* 0 of K satisfying the inequalities CiQHWpiQ
(Q = 1,2,..., q),
|FOc)l
Then
q E Q=l Remark: LeVeque's theorem is the special case q=l,cr=0, X=/i=l of the Assertion (1,1). 9. The next assertions can be put in a rather simpler form, but are somewhat stronger.
Assertion (2,1): Let £(i)*0,..., £( ri )*0 be real algebraic numbers; let £(r +1) * °'*"> ^(r +r ) * ° ^e comP^ex algebraic numbers; and let 5] * 0, for j=l,2,..., r, be a Vj-adic algebraic number. Assume there exists an infinite sequence of distinct elements K*0 of K such that
n Then r ^ 2. Assertion (2,H): Let F(i)(x),...,F(ri+ )(x), Fi(x),...,Fr(x) be arbitrary equal or distinct polynomials with coefficients in K that do not vanish for x=0 and have no multiple factors. Assume there exists an infinite sequence of distinct elements H ^ Q o f K satisfying
n j=r+1
Then T ^ 2.
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LECTURES ON DIOPHANTINE APPROXIMATIONS
10. The results stated so far refer to s'olutions /c*0 which are arbitrary, elements of K. They have analogues in which K =f= 0 is restricted to integers in K, and then stronger assertions can be made. It will suffice to state the analogues to the two assertions (2,1) and (2,n). Assertion (3,1): Let £ (i) * 0,...,$(ri)+ 0 be real algebraic numbers; let 4(r1+l) * 0,..., £(r1+r2)* ° **e complex algebraic numbers; and let |j * 0, for j=l, 2,..., r, be a f^-adic algebraic number. Assume there exists an infinite, sequence of distinct integers K * 0 in K for which i=l Then
w
j=l
rj
]=r+l
rj
T < 1. Assertion (3,H): Let F(i)(x),..., F(r +r j(x), Fi(x),..., Fr(x) be arbitrary equal or distinct polynomials with coefficients in K that do not vanish for x=0 0nrf ifeavg wo multiple factors. Assume there exists an infinite sequence of distinct integers K* 0 in K /or w/wc/z
T ^ 1.
10. In Chapter 9 we gave a number of applications of the Approximation Theorem over the rational field. It is obvious that similar applications can be made of the last assertions.
