Minimal dispersion on the cube and the torus

TL;DR

This paper improves upper bounds for minimal dispersion on the cube and torus by introducing a novel probabilistic lemma that combines random and deterministic point selection, leading to tighter bounds.

Contribution

The paper presents a new probabilistic lemma that enhances upper bounds for minimal dispersion, combining random and non-random point choices.

Findings

Improved upper bounds for minimal dispersion on the cube.

Enhanced bounds for minimal dispersion on the torus.

Introduction of a novel probabilistic lemma for dispersion analysis.

Abstract

We improve some upper bounds for minimal dispersion on the cube and torus. /Our new ingredient is an improvement of a probabilistic lemma used to obtain upper bounds for dispersion in several previous works. Our new lemma combines a random and non-random choice of points in the cube. This leads to better upper bounds for the minimal dispersion.