On the order of magnitude of Sudler products

TL;DR

This paper investigates the asymptotic behavior of Sudler products for quadratic irrationals, especially the golden ratio, establishing formulas, bounds, and a transition point at $a=6$ for continued fraction expansions.

Contribution

It provides the first asymptotic formulas for Sudler products at quadratic irrationals and characterizes the transition point at $a=6$ in their continued fraction expansion.

Findings

Asymptotic formula for Sudler products at the golden ratio.

Boundedness of $rac{P_N(rac{ ext{golden ratio}}{N})}{N}$ as $N o

Transition point at $a=6$ in continued fraction expansion.

Abstract

Given an irrational number $\alpha\in(0,1)$, the Sudler product is defined by $P_N(\alpha) = \prod_{r=1}^{N}2|\sin\pi r\alpha|$. Answering a question of Grepstad, Kaltenb\"ock and Neum\"uller we prove an asymptotic formula for distorted Sudler products when $\alpha$ is the golden ratio $(\sqrt{5}+1)/2$ and establish that in this case $\limsup_{N \to \infty} P_N(\alpha)/N < \infty$. We obtain similar results for quadratic irrationals $\alpha$ with continued fraction expansion $\alpha = [a,a,a,\dots]$ for some integer $a \geq 1$, and give a full characterization of the values of $a$ for which $\liminf_{N \to \infty} P_N(\alpha)>0$ and $\limsup_{N \to \infty} P_N(\alpha) / N < \infty$ hold, respectively. We establish that there is a (sharp) transition point at $a=6$, and resolve as a by-product a problem of the first named author, Larcher, Pillichshammer, Saad Eddin, and Tichy.