## 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.## References

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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