The order of magnitude of Sudler products

Agamemnon Zafeiropoulos (NTNU)

18-Jan-2022, 13:30-14:30 (2 years ago)

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

( paper | slides )


One World Numeration seminar

Series comments: Description: Online seminar on numeration systems and related topics

For questions or subscribing to the mailing list, contact the organisers at numeration@irif.fr

Organizers: Shigeki Akiyama, Ayreena Bakhtawar, Karma Dajani, Kevin Hare, Hajime Kaneko, Niels Langeveld, Lingmin Liao, Wolfgang Steiner*
*contact for this listing

Export talk to