Submit progress

on Value Distribution of Entire Functions with Fabry Gaps

The statement

An entire function is a complex-valued function that is holomorphic on the entire complex plane. A sequence of natural numbers $n : \mathbb{N} \to \mathbb{N}$ is said to have Fabry gaps if $\displaystyle\lim_{k \to \infty} \frac{n_k}{k} = \infty$ . Let $f : \mathbb{C} \to \mathbb{C}$ be an entire function with power series expansion $\displaystyle f(z) = \sum_{k=0}^{\infty} a_k z^{n_k}$ where the coefficients $a_k \in \mathbb{C}$ are all nonzero and the sequence of exponents $(n_k)$ has Fabry gaps, meaning the series converges for all $z \in \mathbb{C}$ . Does $f$ assume every complex value infinitely often? That is, for every $z \in \mathbb{C}$ , is the set $\{x \in \mathbb{C} : f(x) = z\}$ infinite?

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.