Refinement of Macintyre — Evgrafov type theorems

The study of the asymptotic behavior of an entire transcendental function of the form $f(z)= \sum_n a_n z^{p_n}$, $p_n \in \mathbb{N}$, on curves $\gamma$ going to infinity arbitrarily, is a classical problem, goes back to the works of Hadamard, Littlewood and Polia. Polia posed the following problem: under what conditions on $p_n$ does there unbounded sequence $\{ \xi_n \} \subset \gamma$ exist such that $\ln M_f (|\xi_n|) \sim \ln |f(\xi_n )|$ for $\xi_n \to \infty$ (Polya's problem). Here $M_f(r)$ is the maximum of the modulus $f$ on a circle of radius $r$. He showed that if the sequence $\{ p_n \}$ has zero density and $f$ is of finite order, then the indicated relation between $\ln M_f (| \xi_n |)$ and $\ln |f(\xi_n )|$ is always present. This assertion is also true in the case when $f$ has a finite lower order: the final results for this case were obtained by A.M. Gaisin, I.D. Latypov and N.N. Yusupova-Aitkuzhina. We consider the situation when the lower order is equal to infinity. A.M. Gaisin received an answer to Polia's problem in 2003. This is the criterion. If the conditions of this criterion are satisfied not by the sequence $\{ p_n \}$ itself, but only by a subsequence — a sequence of central exponents, then the logarithms of the maximum modulus and modulus of the sum of the series will also be equivalent in the indicated sense on any curve $\gamma $ going to infinity.