The order of magnitude of Sudler products
Agamemnon Zafeiropoulos (NTNU)
Abstract: Given an irrational $\alpha \in [0,1] \smallsetminus \mathbb{Q}$, we define the corresponding Sudler product by $$ P_N(\alpha) = \prod_{n=1}^{N}2|\sin (\pi n \alpha)|. $$ In joint work with C. Aistleitner and N. Technau, we show that when $\alpha = [0;b,b,b…]$ is a quadratic irrational with all partial quotients in its continued fraction expansion equal to some integer b, the following hold:
- If $b\leq 5$, then $\liminf_{N\to \infty}P_N(\alpha) >0$ and $\limsup_{N\to \infty} P_N(\alpha)/N < \infty$.
-If $b\geq 6$, then $\liminf_{N\to \infty}P_N(\alpha) = 0$ and $\limsup_{N\to \infty} P_N(\alpha)/N = \infty$.
We also present an analogue of the previous result for arbitrary quadratic irrationals (joint work with S. Grepstad and M. Neumueller).
dynamical systemsnumber theory
Audience: researchers in the topic
Series comments: Description: Online seminar on numeration systems and related topics
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Organizers: | Shigeki Akiyama, Ayreena Bakhtawar, Karma Dajani, Kevin Hare, Hajime Kaneko, Niels Langeveld, Lingmin Liao, Wolfgang Steiner* |
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