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arXiv · 1801.09416

Asymptotic behaviour of the Sudler product of sines for quadratic irrationals

Sigrid Grepstad·Mario Neumüller
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Abstract

We study the asymptotic behaviour of the sequence of sine products $P_n(\alpha) = \prod_{r=1}^n |2\sin \pi r \alpha|$ for real quadratic irrationals $\alpha$. In particular, we study the subsequence $Q_n(\alpha)=\prod_{r=1}^{q_n} |2\sin \pi r \alpha|$, where $q_n$ is the $n$th best approximation denominator of $\alpha$, and show that this subsequence converges to a periodic sequence whose period equals that of the continued fraction expansion of $\alpha$. This verifies a conjecture recently posed by Mestel and Verschueren.

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Sigrid Grepstad, Mario Neumüller. 2018-01-29. Asymptotic behaviour of the Sudler product of sines for quadratic irrationals. https://arxiv.org/abs/1801.09416

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