On the asymptotic behavior of Sudler products along subsequences

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 Neumüller (see arXiv:1801.09416[math.NT]), are generalized to the case of irrationals with bounded continued fraction coefficients.