  Given an irrational number $\alpha$, we study the asymptotic behaviour of the
Sudler product denoted by $P_N(\alpha) = \prod_{r=1}^N 2\lvert \sin \pi r
\alpha \rvert$. 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 3
only finitely often, which confirms a conjecture of the second-named author and
partially answers a question of J. Shallit. Furthermore, we show that the
Hausdorff dimension of the set of those $\alpha$ that satisfy $\limsup_{N \to
\infty} P_N(\alpha)/N < \infty,\liminf_{N \to \infty} P_N(\alpha) >0$ lies
between $0.7056$ and $0.8677$, which makes significant progress in a question
raised by Aistleitner, Technau, and Zafeiropoulos. We also show that the set of
such $\alpha$ is invariant under the Gauss map $T$.
