An approximate solution to Erdős' maximum modulus points problem

In this note we investigate the asymptotic behavior of the number of maximum modulus points, of an entire function, sitting in a disc of radius r. In 1964, Erdos asked whether there exists a non-monomial function so that this quantity is unbounded? tends to infinity? In 1968 Herzog and Piranian constructed an entire map for which it is unbounded. Nevertheless, it is still unknown today whether it is possible that it tends to infinity or not. In this paper, we construct a transcendental entire function that is arbitrarily close to satisfying this property, thereby giving the strongest evidence supporting a positive answer to this question.(c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http:// creativecommons .org /licenses /by -nc -nd /4 .0/).

Automated summary

This note investigates the asymptotic behavior of the number of maximum modulus points of an entire function in a disc of radius r. While Herzog and Piranian constructed an entire function for which this quantity is unbounded, it is still unknown whether it is possible for it to tend to infinity. In this paper, a transcendental entire function is constructed that is arbitrarily close to satisfying this property, providing strong evidence for a positive answer to this question.