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 Lebesguealmost every $\alpha$, we can prove more: we show that for every nondecreasing 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 

Event title  One World Numeration Seminar 
Event type  Seminar 
Location  Paris, FranceShow on map 
Degree of Recognition  International 
Keywords
 Diophantine approximation
 Continued fractions
 Trigonometric product
Related content

Publications

On extreme values for the Sudler product of quadratic irrationals
Research output: Contribution to journal › Article › peerreview