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

TL;DR

This paper proves that for the golden ratio, the infinite product of sine terms remains bounded away from zero, providing a counterexample to a long-standing question about the behavior of such products for irrationals.

Contribution

It establishes a positive lower bound for the product involving the golden ratio, countering previous conjectures that the limit infimum is always zero for irrationals.

Findings

The product's limit infimum for the golden ratio is strictly positive.

Counterexample to the conjecture that the product tends to zero for all irrationals.

Provides new insights into the distribution of sine products related to irrational numbers.

Abstract

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.