Metric density results for the value distribution of Sudler products

TL;DR

This paper investigates the distribution of Sudler products for almost all irrationals, establishing density results related to their logarithmic growth and answering a question posed by Bence Borda.

Contribution

It provides new density results for the value distribution of Sudler products, including conditions under which certain growth bounds hold with high density, and addresses an open question.

Findings

Sets where $ ext{log} P_N( ext{alpha})$ is below a threshold have upper density 1.

Sets where $ ext{log} P_N( ext{alpha})$ exceeds a threshold have upper density at least 1/2.

Conditions on $ ext{psi}$ determine the density and equality cases for Sudler product growth.

Abstract

We study the value distribution of the Sudler product $P_N(\alpha) := \prod_{n=1}^{N}\lvert2\sin(\pi n \alpha)\rvert$ for Lebesgue-almost every irrational $\alpha$. We show that for every non-decreasing function $\psi: (0,\infty) \to (0,\infty)$ with $\sum_{k=1}^{\infty} \frac{1}{\psi(k)} = \infty$, the set $\{N \in \mathbb{N}: \log P_N(\alpha) \leq -\psi(\log N)\}$ has upper density $1$, which answers a question of Bence Borda. On the other hand, we prove that $\{N \in \mathbb{N}: \log P_N(\alpha) \geq \psi(\log N)\}$ has upper density at least $\frac{1}{2}$, with remarkable equality if $\liminf_{k \to \infty} \psi(k)/(k \log k) \geq C$ for some sufficiently large $C > 0$.