Asymptotic behaviour of the Sudler product of sines for quadratic irrationals

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.