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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
Abstract

There is an extensive literature on the asymptotic order of Sudler’s trigonometric product PN(α) = n=1N|2sin (πnα)| for fixed or for “typical” values of α. We establish a structural result which for a given α characterizes those N for which PN(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 1NMPN(α) and for N=1MPN(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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