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 |
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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
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Publications
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On extreme values for the Sudler product of quadratic irrationals
Research output: Contribution to journal › Article › peer-review