Lebesgue measure of escaping sets of entire functions of completely regular growth

TL;DR

This paper establishes conditions under which the Julia and escaping sets of certain entire functions have positive Lebesgue measure, with applications to solutions of linear differential equations.

Contribution

It provides new criteria linking the growth and zero distribution of entire functions to the measure of their Julia and escaping sets.

Findings

Julia and escaping sets have positive Lebesgue measure under specified conditions.

Conditions relate the indicator function and zero distribution of entire functions.

Applications include solutions to linear differential equations.

Abstract

We give conditions ensuring that the Julia set and the escaping set of an entire function of completely regular growth have positive Lebesgue measure. The essential hypotheses are that the indicator is positive except perhaps at isolated points and that most zeros are located in neighborhoods of finitely many rays. We apply the result to solutions of linear differential equations.