Maximizing Sudler products via Ostrowski expansions and cotangent sums

Aistleitner, Christoph; Borda, Bence

doi:10.2140/ant.2023.17.667

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Maximizing Sudler products via Ostrowski expansions and cotangent sums

Christoph Aistleitner and Bence Borda

Vol. 17 (2023), No. 3, 667–717

DOI: 10.2140/ant.2023.17.667

Abstract

There is an extensive literature on the asymptotic order of Sudler’s trigonometric product $P_{N} (α) = \prod_{n = 1}^{N} | 2 \sin (π n α) |$ for fixed or for “typical” values of $α$ . We establish a structural result which for a given $α$ characterizes those $N$ for which $P_{N} (α)$ attains particularly large values. This characterization relies on the coefficients of $N$ in its Ostrowski expansion with respect to $α$ , and allows us to obtain very precise estimates for $\max_{1 \leq N \leq M} P_{N} (α)$ and for $\sum_{N = 1}^{M} P_{N} {(α)}^{c}$ in terms of $M$ , for any $c > 0$ . Furthermore, our arguments give a natural explanation of the fact that the value of the hyperbolic volume of the complement of the figure-eight knot appears generically in results on the asymptotic order of the Sudler product and of the Kashaev invariant.

Keywords

continued fractions, Ostrowski expansion, cotangent sum, quadratic irrationals, Sudler product, Kashaev invariant

Mathematical Subject Classification

Primary: 11A63, 11J70, 11L03, 26D05

Milestones

Received: 18 March 2021

Revised: 18 January 2022

Accepted: 10 May 2022

Published: 12 April 2023

Authors

Christoph Aistleitner
Institute of Analysis and Number Theory Graz University of Technology Graz Austria

Bence Borda
Institute of Analysis and Number Theory Graz University of Technology Graz Austria	Alfréd Rényi Institute of Mathematics Budapest Hungary

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Vol. 17, No. 3, 2023