On the asymptotic behaviour of Sudler products for badly approximable numbers

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Abstract

Given a badly approximable number $\alpha$, we study the asymptotic behaviour of the Sudler product defined by $P_N(\alpha) = \prod_{r=1}^N 2 | \sin \pi r \alpha |$. We show that $\liminf_{N \to \infty} P_N(\alpha) = 0$ and $\limsup_{N \to \infty} P_N(\alpha)/N = \infty$ whenever the sequence of partial quotients in the continued fraction expansion of $\alpha$ exceeds $7$ infinitely often. This improves results obtained by Lubinsky for the general case, and by Grepstad, Neum\"uller and Zafeiropoulos for the special case of quadratic irrationals. Furthermore, we prove that this threshold value $7$ is optimal, even when restricting $\alpha$ to be a quadratic irrational, which gives a negative answer to a question of the latter authors.