Optimal sine and sawtooth inequalities

Abstract

We determine the optimal inequality of the form ${\sum }_{k=1}^m{a}_k\sin kx\le 1$, 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.

arXiv comment19 pages, 6 figures; v2: Omitted the application to algebraic varieties of general type with small volume, in view of better results for that problem in arXiv:2109.13383

Cited by in corpus (1)

• Varieties of general type with doubly exponential asymptotics