  We study the value distribution of the Sudler product $P_N(\alpha) :=
\prod_{n=1}^{N}\lvert2\sin(\pi n \alpha)\rvert$ for Lebesgue-almost every
irrational $\alpha$. We show that for every non-decreasing function $\psi:
(0,\infty) \to (0,\infty)$ with $\sum_{k=1}^{\infty} \frac{1}{\psi(k)} =
\infty$, the set $\{N \in \mathbb{N}: \log P_N(\alpha) \leq -\psi(\log N)\}$
has upper density $1$, which answers a question of Bence Borda. On the other
hand, we prove that $\{N \in \mathbb{N}: \log P_N(\alpha) \geq \psi(\log N)\}$
has upper density at least $\frac{1}{2}$, with remarkable equality if
$\liminf_{k \to \infty} \psi(k)/(k \log k) \geq C$ for some sufficiently large
$C > 0$.
