Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A positive lower bound for $\liminf_{N\to\infty} \prod_{r=1}^N \left|
  2\sin \pi r \varphi \right|$   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.

18
The categories of this paper are: econ
5
The categories of this paper are: gr-qc
25
The categories of this paper are: patt-sol
21
The categories of this paper are: adap-org
33
The categories of this paper are: plasm-ph
12
The categories of this paper are: nucl-th
6
The categories of this paper are: astro-ph
31
The categories of this paper are: ao-sci
17
The categories of this paper are: q-fin
10
The categories of this paper are: q-bio
3
The categories of this paper are: physics
16
The categories of this paper are: stat
1
The categories of this paper are: cs
13
The categories of this paper are: quant-ph
9
The categories of this paper are: nlin
0
The categories of this paper are: hep-ph