Glucksam 2024 approximate solution erdos maximum modulus points

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question_1_1: Erdős's question as the paper states it: for a non-monomial entire function, whether the number of maximum modulus points on the circle of radius r can have infinite limit superior, and whether it can have infinite limit inferior.

theorem_1_2: Glücksam and Pardo-Simón construct an entire function such that, for every positive epsilon, the number of arcs of the circle of radius r on which the modulus exceeds the maximum modulus minus epsilon, and the number of arcs on which it is below epsilon, both tend to infinity with r.

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Glücksam, Adi and Pardo-Simón, Leticia, An approximate solution to Erdős' maximum modulus points problem. J. Math. Anal. Appl. 531 (2024), no. 1, Paper No. 127768, 20 pp. (DOI 10.1016/j.jmaa.2023.127768). Labels and page numbers on the result pages are those of the arXiv version arXiv:2208.11154v2 (26 September 2023).

For an entire $f$ let $v(r)$ count the maximum modulus points of modulus $r$, that is, the points on $|z|=r$ with $|f(z)|=M(|z|)$. Erdős asked in 1964 whether for a non-monomial entire function $v$ can be unbounded (part (a)) and whether it can tend to infinity (part (b)); Herzog and Piranian settled (a) in 1968 by constructing $f$ with $v(n)=n$ for every $n\in\mathbb N$, but, the paper notes, their refinements do not seem to give any control of $v(r)$ for $r\notin\mathbb N$, and (b) was open when the paper was written. The paper's Theorem 1.2, which the authors present as the strongest evidence for a positive answer to (b), gives one entire $f$ such that, for every $\varepsilon>0$, both the number $v(r,\varepsilon)$ of connected components of $\{|z|=r\}$ intersected with $\{|f(z)|>M(|z|)-\varepsilon\}$ and the number $w(r,\varepsilon)$ of components of $\{|z|=r\}$ intersected with $\{|f(z)|<\varepsilon\}$ tend to infinity as $r\to\infty$. The authors do not conclude (b) for their $f$: the approximation leaves open that only a uniformly bounded number of the arcs of the approximate maximum modulus set on $\{|z|=r\}$ contain a maximum modulus point (p. 3). The construction pastes multiples of $\exp(z^{2^n})$ on sectors by Hörmander's solution of the $\bar\partial$-equation, and the function has infinite lower order (Remark 1.3). This is the reference [GlPa24] in problem 1117, whose site commentary cites it as an approximate affirmative analogue of the second question.

Source: https://arxiv.org/abs/2208.11154. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2208.11154), every other right reserved.

Read status: claims checked. Question 1.1, Theorem 1.2 and Remark 1.3 were read clause by clause on pp. 1--3; the proofs of Sections 2--4 were read in outline only.

Contents

• Question 1.1 (Erdős; p. 2): for a non-monomial entire $f$, can $\limsup_{r\to\infty}v(r)=\infty$ (a), and can $\liminf_{r\to\infty}v(r)=\infty$ (b)?
• Theorem 1.2 (p. 2; proof Section 4, pp. 16--21): there is an entire $f$ with $\lim_{r\to\infty}v(r,\varepsilon)=\infty$ and $\lim_{r\to\infty}w(r,\varepsilon)=\infty$ for every $\varepsilon>0$; Remark 1.3 (p. 3) on its infinite lower order is recorded on the same page.

Bears on. #1117: the paper's Question 1.1 states the problem's two questions; its Theorem 1.2 proves an approximate analogue of the second question, with arcs where $|f|>M(r)-\varepsilon$ in place of maximum modulus points, and does not answer it, as the paper says (p. 3).

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.