Optimal sine and sawtooth inequalities

TL;DR

This paper finds the best inequalities involving sums of sine and sawtooth functions that maximize the sum of coefficients while keeping the sum bounded by one, linking these to equidistribution problems on the circle.

Contribution

It provides exact solutions to optimal inequalities for sine and sawtooth functions, connecting these to equidistribution on the unit circle.

Findings

Optimal sine inequalities with maximal coefficient sum

Exact solutions for sawtooth function inequalities

Connection to equidistribution problems

Abstract

We determine the optimal inequality of the form $\sum_{k=1}^m a_k\sin kx\leq 1$, in the sense that $\sum_{k=1}^m a_k$ is maximal. We also solve exactly the analogous problem for the sawtooth (or signed fractional part) function. Equivalently, we solve exactly an optimization problem about equidistribution on the unit circle.