Metric density results for the value distribution of Sudler products

Abstract:

We study the value distribution of the Sudler product $P_N(\alpha ) ≔\prod _{n=1}^{N}\lvert 2\sin (\pi n \alpha )\rvert$ 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 $C > 0$.