For $\alpha \in \mathbb{R}$ and $N \in \mathbb{N}$, the Sudler product at stage $N$ is defined as
$$P_N(\alpha) := \prod_{r=1}^{N} 2 \left\lvert \sin \pi r\alpha \right\rvert.$$
It is known that $\liminf_{N \to \infty}P_N(\alpha)=0$ whenever the sequence of partial quotients in the continued fraction expansion of $\alpha$ contains infinitely many digits greater than $6$. In fact, it was conjectured by Lubinsky that $\liminf$ equals zero for all real numbers. However, it was shown by Verschueren and independently by Grepstad, Kaltenböck and Neumüller that for $\alpha$ equal to the golden ratio $[0;1,1,1,\ldots]$ one has $\liminf_{N \to \infty}P_N(\alpha)>0$. Later, some other counterexamples were found, all of them also were quadratic irrationals. In a joint paper with Manuel Hauke, we show that $\liminf_{N \to \infty} P_N(\alpha) >0$ whenever the sequence of partial quotients in the continued fraction expansion of $\alpha$ exceeds $3$ only finitely often. Furthermore we deduce a non-trivial lower bound of the HD of the set of $\alpha$ satisfying $\liminf_{N \to \infty} P_N(\alpha) >0$.
