On Lacunary Power Series

271

MA THEMATICS: R. E. A. C. PALEY

VOL. 19, 1933

according to (1). But the number of tori is infinite, and they are mutually external; consequently ir

7r

7r

Area of S > - + -4 +...+-4 +... or

Area of S= + o. In my solution of the problem of Plateau,2 I have shown that every Jordan curve is the boundary of a simply connected minimal surface M: In

xi =

RFi(w),

2 Fj2(w)

=

0.

$=1

This applies to the present example J, where the area of M like that of every other simply connected surface bounded by J, is infinite. A good sense in which M still has the least area property, in spite of the area functional becoming identically infinite, was given by me in a previous paper.3 1 "A Step-Polygon of a Denumerable Infinity of Sides Which Bounds No Finite Area," these PROCEEDINGS, 19, 188-191 (1933). 2 "Solution of the Problem of Plateau," Trans. Amer. Math. Soc., 33, No. 1, 263321 (1931). 3"The Least Area Property of the Minimal Surface Determined by an Arbitrary Jordan Contour," these PROCEEDINGS, 17, 211-216 (1931).

ON LA CUNAR Y POWER SERIES By R. E. A. C. PALEY' DEPARTMENT OF MATHEMATICS, MASSACHUSETTS INSTITUTE OF TECHNOLOGY

Communicated January 14, 1933

1. We consider a power series (1)

ak k

=

1

where nk + 1/nk 2 q > 1 (k = 1, 2, . . .), the circle of convergence is supposed to be z = 1, and k

E 1 |aki= =

.

(2)

I have proved the following theorem, which was suggested to me by

Professor Zygmund.

272

MA THEMA TICS: J. W. ALEXANDER

PROC. N. A. S.

THEOREM I. If the condition (2) is satisfied and a. )- 0, then the series (1) uill converge to any arbitrary complex number in an everywhere dense set of points on the circle of convergence. 2. It seems plausible that the following associated theorem is also true, but so far neither Professor Zygmund nor I have succeeded in proving it. THEOREM II. If the condition (2) is satisfied, then the series (1) will converge to any arbitrary complex number in at least one interior point of the circle of convergence. 3. We observe in passing that Theorem I assures us that the function (1) assumes inside the circle values arbitrarily near to any assigned number, so that the set of exceptional values is at any rate nowhere dense. 1 INTERNATIONAL RESEARCH FELLOW and Research Fellow, Harvard University.

A MATRJX KNOT INVARIANT BY J. W. ALEXANDER DEPARTMENT OF MATHEMATICS, PRINCETON UNIVERSITY

Communicated January 11, 1933

Let K be any smooth, closed curve in a Euclidean 3-space R. Then the curve K determines an infinitely sheeted covering space R. spread over the space R and such that as we go around the curve K we traverse the sheets of R,, in cyclical order. The topological invariants of the space R. are, obviously, functions of the knot type of the curve K. We shall fix our attention, more particularly, on the group G formed by the one-cycles of R. that are independent with respect to homologies. Now, the homology group G cannot, in general, be generated by a finite nujmber of cycles, in the restricted sense of the word "generate." Let us, however, introduce an operator x and form the ring 2 consisting

of all finize sums of the form Eaixi,

(i=O, |

2,...),

where the coefficients as are arbitrary integers. Then, it can be shown that there is a finite set of one-cycles,

ci

l ..(i =1,2,...s)

of the covenng space R. such that every cycle C of R. is homologous to a linear combination of the cycles Ci,