The asymptotic behaviour of Sudler products

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.
Period29 Nov 2022
Event titleOne World Numeration Seminar
Event typeSeminar
LocationParis, FranceShow on map
Degree of RecognitionInternational

Keywords

  • Diophantine approximation
  • Continued fractions
  • Trigonometric product