A positive lower bound for $\liminf_{N\to\infty} \prod_{r=1}^N \left| 2\sin \pi r \varphi \right|$

Abstract

Nearly 60 years ago, Erd\H{o}s and Szekeres raised the question of whether $$\liminf_{N\to \infty} \prod_{r=1}^N \left| 2\sin \pi r \alpha \right| =0$$ 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 $\varphi=(\sqrt{5}-1)/2$, $$\liminf_{N\to \infty} \prod_{r=1}^N \left| 2\sin \pi r \varphi \right| >0 ,$$ providing a negative answer to this long-standing open problem.