The asymptotic behaviour of Sudler products

Abstract: Given an irrational number $\alpha$, we study the asymptotic behaviour of the Sudler product defined by $P_N(\alpha) = \prod_{r=1}^N 2 \lvert \sin \pi r \alpha \rvert$, which appears in many different areas of mathematics. In this talk, we explain the connection between the size of $P_N(\alpha)$ and the Ostrowski expansion of $N$ with respect to $\alpha$. We show that $\liminf_{N \to \infty} P_N(\alpha) = 0$ and $\limsup_{N \to \infty} P_N(\alpha)/N = \infty$, whenever the sequence of partial quotients in the continued fraction expansion of $\alpha$ exceeds $7$ infinitely often, and show that the value $7$ is optimal.

For Lebesgue-almost every $\alpha$, we can prove more: we show that for every non-decreasing function $\psi: (0,\infty) \to (0,\infty)$ with $\sum_{k=1}^{\infty} \frac{1}{\psi(k)} = \infty$ and $\liminf_{k \to \infty} \psi(k)/(k \log k)$ sufficiently large, the conditions $\log P_N(\alpha) \leq -\psi(\log N)$, $\log P_N(\alpha) \geq \psi(\log N)$ hold on sets of upper density $1$ respectively $1/2$.

dynamical systemsnumber theory

Audience: researchers in the topic

( paper | slides )

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One World Numeration seminar

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