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, 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.