Metric density results for the value distribution of Sudler products

Abstract

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