# The asymptotic behaviour of Sudler products

• Manuel Hauke (Speaker)

Activity: Talk or presentationTalk at workshop, seminar or courseScience to science

## Description

Given an irrational number $\alpha$, we study the asymptotic behaviour of the Sudler product defined by $P_N(\alpha) = \prod_{r=1}^N 2 |\sin(\pi r \alpha)|, which appears in many different areas of mathematics. In this talk, we explain the connection between the size of$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} 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.
Period 29 Nov 2022 One World Numeration Seminar Seminar Paris, FranceShow on map International

## Keywords

• Diophantine approximation
• Continued fractions
• Trigonometric product