A note on minimal dispersion of point sets in the unit cube

TL;DR

This paper investigates the minimal number of points needed in a d-dimensional unit cube to intersect all axis-aligned boxes of volume greater than r, establishing bounds that determine their growth rate.

Contribution

It provides an upper bound on N(r,d) that matches previous lower bounds, fully characterizing the growth rate for fixed r in (0,1).

Findings

Established an upper bound on N(r,d).

Matched the bound with existing lower bounds up to a constant.

Determined the growth rate of N(r,d) for fixed r.

Abstract

We study the dispersion of a point set, a notion closely related to the discrepancy. Given a real $r\in (0,1)$ and an integer $d\geq 2$, let $N(r,d)$ denote the minimum number of points inside the $d$-dimensional unit cube $[0,1]^d$ such that they intersect every axis-aligned box inside $[0,1]^d$ of volume greater than $r$. We prove an upper bound on $N(r,d)$, matching a lower bound of Aistleitner et al. up to a multiplicative constant depending only on $r$. This fully determines the rate of growth of $N(r,d)$ if $r\in(0,1)$ is fixed.