A Density Theorem with an Application to Gap Power Series(1) By

Let N be a set of positive integers and let F(z) = I Anz" be an entire function for which An0 (n 0 N). It is reasonable to expect that, if D denotes the density of the set N in some sense, then F(z) will behave somewhat similarly in every angle of opening greater than 2vD. For functions of finite order, the appropriate density seems to be the P6lya maximum density 9. In this paper we introduce a new density 9 which is perhaps the appropriate density for the consideration of functions of unrestricted growth. It is shown that, if III > 2i9, then log M(r) log M(r, I) outside a small exceptional set. Here M(r) denotes the maximum modulus of F(z) on the circle lzl =r and M(r, I) that of F(ret0) for values of 0 in the closed interval I. The method used is closely connected with the question of approximating to functions on an interval by means of linear combinations of the exponentials etx" (n E N).