Minimal dispersion of large volume boxes in the cube

TL;DR

This paper introduces a new construction that improves the upper bounds on the minimal dispersion of large volume boxes in the unit cube, particularly for volumes between 1/4 and 1/2, with bounds depending on the dimension.

Contribution

The authors present a novel construction that tightens the upper bounds on the inverse minimal dispersion for large volume boxes in high-dimensional cubes.

Findings

Improved upper bound on N(r,d) for large volume regimes.

Construction achieves bounds proportional to 1/√(r - 1/4).

Results are sharp for dimensions above a certain constant C_r.

Abstract

In this note we present a construction which improves the best known bound on the minimal dispersion of large volume boxes in the unit cube. Let $d>1$. The dispersion of $T \subset [0,1]^d$ is defined as the supremum of the volume taken over all axis parallel boxes in the cube which do not intersect $T$. The minimal dispersion of $n$ points in the cube is defined as the infimum of the dispersion taken over all $T$ such that $|T| = n$. Define the "large volume" regime as the class of all volumes $\frac{1}{4} < r \leq \frac{1}{2}$. The inverse of the minimal dispersion is denoted as $N(r,d)$. When the volume is large, the best known upper bound on $N(r,d)$ is of the order $ {(r - \frac{1}{4})}^{-1}.$ The construction presented in this note yields an upper bound given by $$N(r,d) \leq \left \lfloor \frac{\pi}{\sqrt{r - \frac{1}{4}}} \right \rfloor - 3 .$$ Some of our intermediate estimates are sharp given the condition that $d \geq C_{r}$, where $C_r$ is a positive constant which depends only on the volume $r$.