A density theorem with an application to gap power series

Abstract

Let N be a set of positive integers and let \[ F ( z ) = ∑ A n z n F(z) = \sum {{A_n}{z^n}} \] be an entire function for which A n = 0 ( n ∉ N ) {A_n} = 0(n \notin N) . It is reasonable to expect that, if D denotes the density of the set N in some sense, then F ( z ) F(z) will behave somewhat similarly in every angle of opening greater than 2 π D 2\pi D . For functions of finite order, the appropriate density seems to be the Pólya maximum density P \mathcal {P} . In this paper we introduce a new density D \mathcal {D} which is perhaps the appropriate density for the consideration of functions of unrestricted growth. It is shown that, if | I | > 2 π D |I| > 2\pi \mathcal {D} , then \[ log ⁡ M ( r ) ∼ log ⁡ M ( r , I ) \log M(r) \sim \log M(r,I) \] outside a small exceptional set. Here M ( r ) M(r) denotes the maximum modulus of F ( z ) F(z) on the circle | z | = r |z| = r and M ( r , I ) M(r,I) that of F ( r e i θ ) F(r{e^{i\theta }}) for values of θ \theta 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 i x n ( n ∈ N ) {e^{ixn}}(n \in N) .