A positive lower bound for ${\mathrm{liminf}}_{N\to \infty }{\prod }_{r=1}^N\left|2\sin \pi r\phi \right|$

Abstract

Nearly 60 years ago, Erdős and Szekeres raised the question of whether $$\underset{N\to \infty }{\mathrm{liminf}}\prod _{r=1}^N\left|2\sin \pi r\alpha \right|=0$$ 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 it was suggested that the answer is yes in general. However, we show in this paper that for the golden ratio $\phi =(\sqrt{5}-1)/2$, $$\underset{N\to \infty }{\mathrm{liminf}}\prod _{r=1}^N\left|2\sin \pi r\phi \right|>0,$$ providing a negative answer to this long-standing open problem.

We were recently made aware that a first proof of our main result is given in Paul Verschueren's PhD thesis from 2016. An addendum and citation have been added