Asymptotic behaviour of the Sudler product of sines for quadratic irrationals

TL;DR

This paper investigates the asymptotic behavior of sine products for quadratic irrationals, proving that specific subsequences converge to periodic sequences aligned with continued fraction periods, confirming a recent conjecture.

Contribution

It establishes the convergence of sine product subsequences for quadratic irrationals to periodic sequences matching their continued fraction periods, confirming a conjecture by Mestel and Verschueren.

Findings

Subsequences of sine products converge to periodic sequences.

The period of the limit sequence matches the continued fraction period.

The conjecture by Mestel and Verschueren is verified.

Abstract

We study the asymptotic behaviour of the sequence of sine products $P_n(\alpha) = \prod_{r=1}^n |2\sin \pi r \alpha|$ for real quadratic irrationals $\alpha$. In particular, we study the subsequence $Q_n(\alpha)=\prod_{r=1}^{q_n} |2\sin \pi r \alpha|$, where $q_n$ is the $n$th best approximation denominator of $\alpha$, and show that this subsequence converges to a periodic sequence whose period equals that of the continued fraction expansion of $\alpha$. This verifies a conjecture recently posed by Mestel and Verschueren.