New bounds on the minimal dispersion

TL;DR

This paper introduces a new construction method for approximating axis-parallel boxes in the unit cube, leading to improved bounds on the minimal dispersion of point sets, with applications to k-dispersion.

Contribution

It provides novel bounds on minimal dispersion using a new construction, improving upon previous results in both periodic and non-periodic settings.

Findings

Improved upper bounds for minimal dispersion in certain regimes

Bounds are sharp up to a double logarithmic factor

Application of construction to k-dispersion

Abstract

We provide a new construction for a set of boxes approximating axis-parallel boxes of fixed volume in $[0, 1]^d$. This improves upper bounds for the minimal dispersion of a point set in the unit cube and its inverse in both the periodic and non-periodic settings in certain regimes. Up to double logarithmic factor, our bounds are sharp. We also apply our construction to $k$-dispersion.