  There is an extensive literature on the asymptotic order of Sudler's
trigonometric product $P_N (\alpha) = \prod_{n=1}^N |2 \sin (\pi n \alpha)|$
for fixed or for "typical" values of $\alpha$. In the present paper we
establish a structural result, which for a given $\alpha$ characterizes those
$N$ for which $P_N(\alpha)$ attains particularly large values. This
characterization relies on the coefficients of $N$ in its Ostrowski expansion
with respect to $\alpha$, and allows us to obtain very precise estimates for
$\max_{1 \le N \leq M} P_N(\alpha)$ and for $\sum_{N=1}^M P_N(\alpha)^c$ in
terms of $M$, for any $c>0$. Furthermore, our arguments give a natural
explanation of the fact that the value of the hyperbolic volume of the
complement of the figure-eight knot appears generically in results on the
asymptotic order of the Sudler product and of the Kashaev invariant.
