On the distribution of Sudler products and Birkhoff sums for the irrational rotation

TL;DR

This work analyzes the value distribution of the Sudler product $P_N(\alpha)=\prod_{n=1}^N|2\sin(\pi n\alpha)|$ and related Diophantine products for various irrationals $\alpha$, revealing a rich dependence on Diophantine properties. By interpreting $\log P_N(\alpha)$ as a Birkhoff sum with $f(x)=\log|2\sin(\pi x)|$ and leveraging its Fourier expansion, the authors derive precise first- and second-moment formulas for broad classes of periodic functions and establish central limit theorems for specific quadratic irrationals, while showing anticoncentration phenomena for almost every $\alpha$. They prove concentration around $N^{1/2}$ for badly approximable $\alpha$, with Gaussian fluctuations in the quadratic case, and they connect Sudler products to the temporal limit behavior of Birkhoff sums, including a conjectural sharpening of Bromberg–Ulcigrai type results. The results extend to the Diophantine product via a quantitative comparison, and to Euler’s number $e$ and almost all $\alpha$, highlighting the delicate interplay between Diophantine approximation and rotational dynamics in value distributions.

Abstract

We study the value distribution of the Sudler product $\prod_{n=1}^N |2 \sin (πn α)|$ and the Diophantine product $\prod_{n=1}^N (2e\| n α\|)$ for various irrational $α$, as $N$ ranges in a long interval of integers. At badly approximable irrationals these products exhibit strong concentration around $N^{1/2}$, and at certain quadratic irrationals they even satisfy a central limit theorem. In contrast, at almost every $α$ we observe an interesting anticoncentration phenomenon when the typical and the extreme values are of the same order of magnitude. Our methods are equally suited for the value distribution of Birkhoff sums $\sum_{n=1}^N f(n α)$ for circle rotations. Using Diophantine approximation and Fourier analysis, we find the first and second moment for an arbitrary periodic $f$ of bounded variation, and (almost) prove a conjecture of Bromberg and Ulcigrai on the appropriate scaling factor in a so-called temporal limit theorem. Birkhoff sums also satisfy a central limit theorem at certain quadratic irrationals.

Theorems & Definitions (17)

• Theorem 1
• Theorem 2
• Theorem 3: Beck BE1BE2BE3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Conjecture
• Theorem 8
• Theorem 9