# Exponential polynomials with Fatou and non-escaping sets of finite Lebesgue measure

## TL;DR

This paper establishes conditions under which exponential polynomials have Fatou and non-escaping sets of finite Lebesgue measure, focusing on growth conditions that control the size of the polynomial outside finite measure sets.

## Contribution

It provides new criteria linking exponential polynomial growth to the finiteness of Lebesgue measure of Fatou and non-escaping sets.

## Key findings

- Fatou set has finite Lebesgue measure under specified growth conditions
- Non-escaping set also has finite Lebesgue measure with these conditions
- Growth condition |f(z)| ≥ exp(|z|^α) outside finite measure sets

## Abstract

We give conditions ensuring that the Fatou set and the complement of the fast escaping set of an exponential polynomial $f$ have finite Lebesgue measure. Essentially, these conditions are designed such that $|f(z)|\ge\exp(|z|^\alpha)$ for some $\alpha>0$ and all $z$ outside a set of finite Lebesgue measure.