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arXiv · 2109.04342

On the order of magnitude of Sudler products II

Sigrid Grepstad·Mario Neumüller·Agamemnon Zafeiropoulos
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Abstract

We study the asymptotic behavior of Sudler products $P_N(\alpha)= \prod_{r=1}^{N}2|\sin \pi r\alpha|$ for quadratic irrationals $\alpha \in \mathbb{R}$. In particular, we verify the convergence of certain perturbed Sudler products along subsequences, and show that $\liminf_N P_N(\alpha) = 0$ and $\limsup_N P_N(\alpha)/N = \infty$ whenever the maximal digit in the continued fraction expansion of $\alpha$ exceeds $23$. This generalizes results obtained for the period one case $\alpha=[0; \overline{a}]$.

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Sigrid Grepstad, Mario Neumüller, Agamemnon Zafeiropoulos. 2021-09-09. On the order of magnitude of Sudler products II. https://arxiv.org/abs/2109.04342

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