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.

To appear in: D. Bilyk, J. Dick, F. Pillichshammer (Eds.) Discrepancy theory, Radon Series on Computational and Applied Mathematics