An approximate solution to Erd\"os' maximum modulus points problem

TL;DR

This paper constructs a transcendental entire function that closely approaches having an unbounded number of maximum modulus points within a disc, providing strong evidence towards Erdős's long-standing open question.

Contribution

The authors develop an approximate solution to Erdős's problem by constructing a transcendental entire function that nearly exhibits unbounded maximum modulus points.

Findings

Constructed a transcendental entire function close to having unbounded maximum modulus points.

Provides evidence supporting the possibility of unbounded maximum modulus points for entire functions.

Advances understanding of the asymptotic behavior of maximum modulus points in complex analysis.

Abstract

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, Erd\Humlaut{o}s 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.