A tight lower bound on the minimal dispersion

2023/11/17 by M. Trödler, Jan Volec, Trödler, Matěj +3 · 1 citation
Mathematics · #FOS: Mathematics #Limits and Structures in Graph Theory #Markov Chains and Monte Carlo Methods #Mathematical Approximation and Integration #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.2311.10666

openalex publication_date 2023/11/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give a new lower bound for the minimal dispersion of a point set in the unit cube and its inverse function in the high dimension regime. This is done by considering only a very small class of test boxes, which allows us to reduce bounding the dispersion to a problem in extremal set theory. Specifically, we translate a lower bound on the size of r-cover-free families to a lower bound on the inverse of the minimal dispersion of a point set. The lower bound we obtain matches the recently obtained upper bound on the minimal dispersion up to logarithmic terms.

Cited by

• Minimal dispersion on the sphere

Related

• A remark on the minimal dispersion
• New bounds on the minimal dispersion
• Lower bounds on the minimal dispersion of point sets via cover-free families
• Minimal dispersion on the cube and the torus
• An Upper Bound of the Minimal Dispersion via Delta Covers