Maximizing Sudler products via Ostrowski expansions and cotangent sums

Abstract

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 ${\max }_{1\le N\le M}{P}_N(\alpha )$ 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.

58 pages