On the order of magnitude of Sudler products II

Abstract

We study the asymptotic behavior of Sudler products $P_N(\alpha )={\prod }_{r=1}^{N}2|\sin \pi r\alpha |$ for quadratic irrationals $\alpha \in \mathbb{R}$. In particular, we verify the convergence of certain perturbed Sudler products along subsequences, and show that ${\mathrm{liminf}}_N{P}_N(\alpha )=0$ and ${\mathrm{limsup}}_N{P}_N(\alpha )/N=\infty$ whenever the maximal digit in the continued fraction expansion of $\alpha$ exceeds $23$. This generalizes results obtained for the period one case $\alpha =[0;\overline{a}]$.

arXiv commentThe second version contains some minor corrections