Measurably entire functions and their growth

TL;DR

This paper investigates the growth rates of measurably entire functions associated with free C-actions on probability spaces, establishing upper bounds on their growth and complementing existing lower bound results.

Contribution

It demonstrates the existence of measurably entire functions with controlled growth rates, specifically not exceeding exp (exp[log^p |z|]) for p > 3, advancing understanding of their growth behavior.

Findings

Existence of measurably entire functions with growth ≤ exp (exp[log^p |z|]) for p > 3

Complementary to prior results showing lower bounds for growth rates

Provides bounds that narrow the possible growth spectrum of such functions

Abstract

In 1997 B. Weiss introduced the notion of measurably entire functions and proved that they exist on every arbitrary free C- action defined on standard probability space. In the same paper he asked about the minimal possible growth of measurably entire functions. In this work we show that for every arbitrary free C- action defined on a standard probability space there exists a measurably entire function whose growth does not exceed exp (exp[log^p |z|]) for any p > 3. This complements a recent result by Buhovski, Gl\"ucksam, Logunov, and Sodin (arXiv:1703.08101) who showed that such functions cannot grow slower than exp (exp[log^p |z|]) for any p < 2.