On the order of magnitude of Sudler products II

TL;DR

This paper investigates the asymptotic behavior of Sudler products for quadratic irrationals, revealing divergence properties and convergence along subsequences, extending previous results for simple continued fractions.

Contribution

It generalizes earlier findings by analyzing Sudler products for quadratic irrationals with complex continued fraction expansions, including divergence and convergence properties.

Findings

 ext{max digit} > 23, ext{ then } ext{liminf} P_N( ext{ extalpha})=0

ext{limsup}_N P_N( ext{ extalpha})/N = ext{infinity}

ext{Certain perturbed products converge along subsequences}

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 $\liminf_N P_N(\alpha) = 0$ and $\limsup_N P_N(\alpha)/N = \infty$ whenever the maximal digit in the continued fraction expansion of $\alpha$ exceeds $23$. This generalizes results obtained for the period one case $\alpha=[0; \overline{a}]$.