TY - JOUR
T1 - An approximate solution to Erdős' maximum modulus points problem
AU - Glücksam, Adi
AU - Pardo-Simón, Leticia
N1 - Funding Information:
This project originated from discussions that followed a talk given by the second author in the Complex Analysis online seminar series “CAvid”. We thank Rod Halburd for organizing it. We are also grateful to Jack Burkart and David Sixsmith for many helpful discussions, and to the referee, whose thoughtful and meticulous report has greatly improved this paper. This material is based upon work supported by the National Science Foundation under Grant No. 1440140 , while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the winter semester of 2022.
Publisher Copyright:
© 2023 The Author(s)
PY - 2024/3/1
Y1 - 2024/3/1
N2 - 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, Erdős 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.
AB - 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, Erdős 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.
KW - Entire functions
KW - Erdős' problem
KW - Maximum modulus
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U2 - 10.1016/j.jmaa.2023.127768
DO - 10.1016/j.jmaa.2023.127768
M3 - Article
AN - SCOPUS:85172184822
SN - 0022-247X
VL - 531
JO - Journal of Mathematical Analysis and Applications
JF - Journal of Mathematical Analysis and Applications
IS - 1
M1 - 127768
ER -