On the asymptotic behaviour of Sudler products for badly approximable numbers

2022/10/14 by Manuel Hauke, Hauke, Manuel · 1 citation
Mathematics · #11L03 #60F05 #Analytic Number Theory Research #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Primary 11J70 #Secondary 11J68

paper · pdf · doi:10.48550/arxiv.2210.07743

openalex publication_date 2022/10/14 · openalex created_date 2022/10/19 · openalex updated_date 2026/07/28

Abstract

Given a badly approximable number α, we study the asymptotic behaviour of the Sudler product defined 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 7 infinitely often. This improves results obtained by Lubinsky for the general case, and by Grepstad, Neumüller and Zafeiropoulos for the special case of quadratic irrationals. Furthermore, we prove that this threshold value 7 is optimal, even when restricting α to be a quadratic irrational, which gives a negative answer to a question of the latter authors.

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