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

Abstract

We study the value distribution of the Sudler product $\prod_{n=1}^N |2 \sin (\pi n \alpha )|$ and the Diophantine product $\prod_{n=1}^N (2e\| n \alpha \|)$ for various irrational $\alpha$, 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 $\alpha$ 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 $\sum_{n=1}^N f(n \alpha )$ 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.