Asymptotic behaviour of the Sudler product of sines for quadratic\n irrationals

2018/01/29 by Sigrid Grepstad, Grepstad, Sigrid, Mario Neumüller +1
Computer Science · Mathematics · #11J70 #41A60 #Analytic Number Theory Research #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.1801.09416

openalex publication_date 2018/01/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the asymptotic behaviour of the sequence of sine products\nP_n(\α) = \∏_r=1ⁿ |2\sin \π r \α| for real quadratic\nirrationals \α. In particular, we study the subsequence\nQ_n(\α)=\∏_r=1^q_n |2\sin \π r \α|, where q_n is the nth\nbest approximation denominator of \α, and show that this subsequence\nconverges to a periodic sequence whose period equals that of the continued\nfraction expansion of \α. This verifies a conjecture recently posed by\nMestel and Verschueren.\n

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