Metric density results for the value distribution of Sudler products

Abstract

We study the value distribution of the Sudler product P N ( α ) ≔ ∏ n = 1 N | 2 sin ⁡ ( π n α ) | P_N(\alpha ) ≔\prod _{n=1}^{N}\lvert 2\sin (\pi n \alpha )\rvert for Lebesgue-almost every irrational α \alpha . We show that for every non-decreasing function ψ : ( 0 , ∞ ) → ( 0 , ∞ ) \psi : (0,\infty ) \to (0,\infty ) with ∑ k = 1 ∞ 1 ψ ( k ) = ∞ \sum _{k=1}^{\infty } \frac {1}{\psi (k)} = \infty , the set { N ∈ N : log ⁡ P N ( α ) ≤ − ψ ( log ⁡ N ) } \{N \in \mathbb {N}: \log P_N(\alpha ) \leq -\psi (\log N)\} has upper density 1 1 , which answers a question of Bence Borda. On the other hand, we prove that { N ∈ N : log ⁡ P N ( α ) ≥ ψ ( log ⁡ N ) } \{N \in \mathbb {N}: \log P_N(\alpha ) \geq \psi (\log N)\} has upper density at least 1 2 \frac {1}{2} , with remarkable equality if lim inf k → ∞ ψ ( k ) / ( k log ⁡ k ) ≥ C \liminf _{k \to \infty } \psi (k)/(k \log k) \geq C for some sufficiently large C > 0 C > 0 .