Metric density results for the value distribution of Sudler products

Abstract

We study the value distribution of the Sudler product $P_N(\alpha ):={\prod }_{n=1}^N|2\sin (\pi n\alpha )|$ 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 $\{N\in \mathbb{N}:\log P_N(\alpha )\le -\psi (\log N)\}$ has upper density $1$, which answers a question of Bence Borda. On the other hand, we prove that $\{N\in \mathbb{N}:\log P_N(\alpha )\ge \psi (\log N)\}$ has upper density at least $\frac{1}{2}$, with remarkable equality if ${\mathrm{liminf}}_{k\to \infty }\psi (k)/(k\log k)\ge C$ for some sufficiently large $C>0$.

13 pages