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(α)=Π_n=1^N|2sin(πnα)| for fixed or for “typical” values of α. We establish a structural result which for a given α characterizes those N for which P_N(α) attains particularly large values. This characterization relies on the coefficients of N in its Ostrowski expansion with respect to α, and allows us to obtain very precise estimates for max_1≤N≤M P_N(α) and for Σ_N=1^M P_N(α)^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.