Maximizing Sudler products via Ostrowski expansions and cotangent sums

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 "typical" values $\alpha$. In present paper we establish a structural result, which given $\alpha$ characterizes those $N$ $P_N(\alpha)$ attains particularly large values. This characterization relies coefficients in its Ostrowski expansion with respect to $\alpha$, and allows us obtain very precise estimates $\max_{1 \le N \leq M} P_N(\alpha)$ $\sum_{N=1}^M P_N(\alpha)^c$ terms $M$, any $c>0$. Furthermore, our arguments give natural explanation fact that value hyperbolic volume complement figure-eight knot appears generically results Sudler Kashaev invariant.