Hausdorff dimension estimates for Sudler products with positive lower bound

Abstract

Given an irrational number $\alpha$, we study the asymptotic behaviour of the Sudler product denoted by $P_N(\alpha) = \prod_{r=1}^N 2\lvert \sin \pi r \alpha \rvert$. We show that $\liminf_{N \to \infty} P_N(\alpha) >0$ and $\limsup_{N \to \infty} P_N(\alpha)/N < \infty$ whenever the sequence of partial quotients in the continued fraction expansion of $\alpha$ exceeds 3 only finitely often, which confirms a conjecture of the second-named author and partially answers a question of J. Shallit. Furthermore, we show that the Hausdorff dimension of the set of those $\alpha$ that satisfy $\limsup_{N \to \infty} P_N(\alpha)/N < \infty,\liminf_{N \to \infty} P_N(\alpha) >0$ lies between $0.7056$ and $0.8677$, which makes significant progress in a question raised by Aistleitner, Technau, and Zafeiropoulos. We also show that the set of such $\alpha$ is invariant under the Gauss map $T$.