Hausdorff dimension estimates for Sudler products with positive lower bound

2023/12/11 by Gayfulin, Dmitry, Hauke, Manuel · 1 citation
#11J70 #26D05 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2312.06548

Abstract

Given an irrational number α, we study the asymptotic behaviour of the Sudler product denoted by P_N(α) = ∏_r=1^N 2| sin πr α|. We show that \liminf_{N → ∞} P_N(α) >0 and \limsup_{N → ∞} P_N(α)/N < ∞ whenever the sequence of partial quotients in the continued fraction expansion of α 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 α that satisfy \limsup_{N → ∞} P_N(α)/N < ∞,\liminf_{N → ∞} P_N(α) >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 α is invariant under the Gauss map T.

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