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A positive lower bound for $ \liminf_{N\rightarrow\infty} \prod_{r=1}^N \Vert 2\sin\pi r \varphi\Vert$


Authors: Sigrid Grepstad, Lisa Kaltenböck and Mario Neumüller
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
Published electronically: May 17, 2019
MathSciNet review: 4011519
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Abstract | References | Similar Articles | Additional Information

Abstract: Nearly 60 years ago, Erdős and Szekeres raised the question of whether

$\displaystyle \liminf _{N\to \infty } \prod _{r=1}^N \left \vert 2\sin \pi r \alpha \right \vert =0$    

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$,

$\displaystyle \liminf _{N\to \infty } \prod _{r=1}^N \left \vert 2\sin \pi r \varphi \right \vert >0 ,$    

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
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

DOI: https://doi.org/10.1090/proc/14611
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
Article copyright: © Copyright 2019 American Mathematical Society