Nearly 60 years ago, Erdős and Szekeres raised the question of whether
lim inf
N
→
∞
∏
r
=
1
N
|
2
sin
π
r
α
|
=
0
\begin{equation*} \liminf _{N\to \infty } \prod _{r=1}^N \left | 2\sin \pi r \alpha \right | =0 \end{equation*}
for all irrationals
α
\alpha
. Despite its simple formulation, the question has remained unanswered. It was shown by Lubinsky in 1999 that the answer is yes if
α
\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
φ
=
(
5
−
1
)
/
2
\varphi =(\sqrt {5}-1)/2
,
lim inf
N
→
∞
∏
r
=
1
N
|
2
sin
π
r
φ
|
>
0
,
\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.