On the asymptotic behaviour of the sine product ∏_r=1ⁿ|2sin(πr α)|

2019/09/03 by Sigrid Grepstad, Grepstad, Sigrid, Lisa Kaltenböck +3
Mathematics · #11B39 (primary) #11K31 (secondary) #11L15 #26D05 #41A60 #Analytic Number Theory Research #FOS: Mathematics #Mathematical functions and polynomials #Mathematics and Applications #Number Theory (math.NT) #math.NT #msc:11B39 #msc:11K31 #msc:11L15 #msc:26D05 #msc:41A60

paper · pdf · doi:10.48550/arxiv.1909.00980

To appear in: D. Bilyk, J. Dick, F. Pillichshammer (Eds.) Discrepancy theory, Radon Series on Computational and Applied Mathematics

arxiv created 2019/09/03 · openalex publication_date 2019/09/03 · arxiv updated 2019/09/04 · openalex created_date 2024/04/10 · openalex updated_date 2026/07/28

Abstract

In this paper we review recently established results on the asymptotic behaviour of the trigonometric product P_n(α) = ∏_r=1ⁿ |2sin πr α| as n→ ∞. We focus on irrationals α whose continued fraction coefficients are bounded. Our main goal is to illustrate that when discussing the regularity of P_n(α), not only the boundedness of the coefficients plays a role; also their size, as well as the structure of the continued fraction expansion of α, is important.

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