Binmore, K.G.; (1970) A density theorem with an application to gap power series. Transactions of the American Mathematical Society , 148 (2) pp. 367-384.

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Abstract

Let N be a set of positive integers and let $F(z)=\Sigma \ A_{n}z^{n}$ be an entire function for which $A_{n}=0\ (n\not\in N)$. 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 2πD. For functions of finite order, the appropriate density seems to be the Pólya maximum density P. In this paper we introduce a new density D which is perhaps the appropriate density for the consideration of functions of unrestricted growth. It is shown that, if $|I|>2\pi \scr{D}$, then ${\rm log}\ M(r)\sim {\rm log}\ M(r,I)$ outside a small exceptional set. Here M(r) denotes the maximum modulus of F(z) on the circle $|z|=r$ and M(r, I) that of $F(re^{i\theta})$ for values of θ 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 $e^{ixn}\ (n\in N)$.

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