Maximizing Sudler products via Ostrowski expansions and cotangent sums

TL;DR

This paper provides a structural characterization of when Sudler's product attains large values, using Ostrowski expansions, and offers precise estimates for its maximum and sums, linking to hyperbolic volume of the figure-eight knot complement.

Contribution

It introduces a novel structural approach based on Ostrowski expansions to analyze Sudler's products and connects these results to hyperbolic geometry and knot theory.

Findings

Characterizes large Sudler products via Ostrowski coefficients.

Provides precise asymptotic estimates for maxima and sums of Sudler products.

Links Sudler product behavior to hyperbolic volume of the figure-eight knot complement.

Abstract

There is an extensive literature on the asymptotic order of Sudler's trigonometric product $P_N (\alpha) = \prod_{n=1}^N |2 \sin (\pi n \alpha)|$ for fixed or for "typical" values of $\alpha$. In the present paper we establish a structural result, which for a given $\alpha$ characterizes those $N$ for which $P_N(\alpha)$ attains particularly large values. This characterization relies on the coefficients of $N$ in its Ostrowski expansion with respect to $\alpha$, and allows us to obtain very precise estimates for $\max_{1 \le N \leq M} P_N(\alpha)$ and for $\sum_{N=1}^M P_N(\alpha)^c$ 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.