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

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.