On the asymptotic behaviour of the sine product $\prod_{r=1}^n|2\sin(\pi r \alpha)|$

TL;DR

This paper reviews recent results on the asymptotic behavior of the sine product for irrationals with bounded continued fraction coefficients, emphasizing the influence of the coefficients' size and structure.

Contribution

It highlights the importance of both the boundedness and the specific structure of continued fraction coefficients in understanding the sine product's asymptotics.

Findings

The asymptotic behavior depends on the size of continued fraction coefficients.

The structure of the continued fraction expansion influences the sine product's regularity.

Bounded coefficients alone do not fully determine the asymptotic properties.

Abstract

In this paper we review recently established results on the asymptotic behaviour of the trigonometric product $P_n(\alpha) = \prod_{r=1}^n |2\sin \pi r \alpha|$ as $n\to \infty$. We focus on irrationals $\alpha$ whose continued fraction coefficients are bounded. Our main goal is to illustrate that when discussing the regularity of $P_n(\alpha)$, not only the boundedness of the coefficients plays a role; also their size, as well as the structure of the continued fraction expansion of $\alpha$, is important.