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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 hasremained unanswered. It was shown by Lubinsky in 1999 that the answer is yes if$\alpha$ has unbounded continued fraction coefficients, and it was suggestedthat the answer is yes in general. However, we show in this paper that for thegolden 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.

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