On the Dispersion of Sparse Grids

TL;DR

This paper introduces a simple construction of point sets in high-dimensional cubes that intersect all large-volume axis-aligned boxes, addressing a fundamental problem in discrepancy theory and geometric combinatorics.

Contribution

The paper presents a new, easy-to-handle point set in high dimensions that guarantees intersection with all large-volume axis-aligned boxes, a property previously known only for more complex sets.

Findings

Constructed point sets intersect all boxes with volume > ε

Sets are simple to handle and implement

Optimal or near-optimal size for given parameters

Abstract

For any natural number $d$ and positive number $\varepsilon$, we present a point set in the $d$-dimensional unit cube $[0,1]^d$ that intersects every axis-aligned box of volume greater than $\varepsilon$. These point sets are very easy to handle and in a vast range for $\varepsilon$ and $d$, we do not know any smaller set with this property.