On the asymptotic behavior of Sudler products along subsequences

TL;DR

This paper studies the asymptotic behavior of Sudler products along subsequences related to best rational approximations of irrational numbers, providing bounds and generalizing convergence results for irrationals with bounded continued fractions.

Contribution

It extends existing convergence results for Sudler products to irrationals with bounded continued fraction coefficients and establishes bounds for their subsequences.

Findings

Derived upper and lower bounds for subsequences of Sudler products.

Generalized convergence results to a broader class of irrationals.

Connected asymptotic behavior to continued fraction properties.

Abstract

Let $\alpha \in (0,1)$ and irrational. We investigate the asymptotic behaviour of sequences of certain trigonometric products (Sudler products) $(P_N(\alpha))_{N\in\mathbb{N}}$ with $$P_N(\alpha) =\prod_{r=1}^N|2\sin(\pi r \alpha)|.$$ More precisely, we are interested in the asymptotic behaviour of subsequences of the form $(P_{q_n(\alpha)}(\alpha))_{n\in\mathbb{N}}$, where $q_n(\alpha)$ is the $n$th best approximation denominator of $\alpha$. Interesting upper and lower bounds for the growth of these subsequences are given, and convergence results, obtained by Mestel and Verschueren (see arXiv:1411.2252math[DS]) and Grepstad and Neum\"uller (see arXiv:1801.09416[math.NT]), are generalized to the case of irrationals with bounded continued fraction coefficients.