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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A positive lower bound for $\liminf _{N\to \infty } \prod _{r=1}^N \left | 2\sin \pi r \varphi \right |$
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by Sigrid Grepstad, Lisa Kaltenböck and Mario Neumüller PDF
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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Additional Information
  • Sigrid Grepstad
  • Affiliation: Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway
  • MR Author ID: 1047019
  • Email: sigrid.grepstad@ntnu.no
  • 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
  • © Copyright 2019 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 147 (2019), 4863-4876
  • MSC (2010): Primary 26D05, 41A60, 11B39; Secondary 11L15, 11K31
  • DOI: https://doi.org/10.1090/proc/14611
  • MathSciNet review: 4011519