  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.
