Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

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A positive lower bound for $\liminf _{N\to \infty } \prod _{r=1}^N \left | 2\sin \pi r \varphi \right |$HTML articles powered by AMS MathViewer

by Sigrid Grepstad, Lisa Kaltenböck and Mario Neumüller
Proc. Amer. Math. Soc. 147 (2019), 4863-4876 Request permission

Abstract:

Nearly 60 years ago, Erdős and Szekeres raised the question of whether \begin{equation*} \liminf _{N\to \infty } \prod _{r=1}^N \left | 2\sin \pi r \alpha \right | =0 \end{equation*} for all irrationals $\alpha$. Despite its simple formulation, the question has remained unanswered. It was shown by Lubinsky in 1999 that the answer is yes if $\alpha$ has unbounded continued fraction coefficients, and he suggested that the answer is yes in general. However, we show in this paper that for the golden ratio $\varphi =(\sqrt {5}-1)/2$, \begin{equation*} \liminf _{N\to \infty } \prod _{r=1}^N \left | 2\sin \pi r \varphi \right | >0 , \end{equation*} providing a negative answer to this long-standing open problem.
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• Affiliation: Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway
• MR Author ID: 1047019
• Lisa Kaltenböck
• Affiliation: Department of Financial Mathematics and Applied Number Theory, Johannes Kepler University, Altenbergerstraße 69, 4040 Linz, Austria
• Email: lisa.kaltenboeck@jku.at
• Mario Neumüller
• Affiliation: Department of Financial Mathematics and Applied Number Theory, Johannes Kepler University, Altenbergerstraße 69, 4040 Linz, Austria
• Email: mario.neumueller@jku.at
• Received by editor(s): October 8, 2018
• Received by editor(s) in revised form: February 14, 2019
• Published electronically: May 17, 2019
• Additional Notes: The first author was supported in part by Grant 275113 of the Research Council of Norway.
The second and third authors were funded by the Austrian Science Fund (FWF): Project F5507-N26 and Project F5509-N26, which were part of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications”.
• Communicated by: Amanda Folsom