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A tight lower bound on the minimal dispersion

Matěj Trödler, Jan Volec, Jan Vybíral

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.

A tight lower bound on the minimal dispersion

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 -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.
Paper Structure (2 sections, 3 theorems, 15 equations)

This paper contains 2 sections, 3 theorems, 15 equations.

Table of Contents

  1. Introduction and the Main Result
  2. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1

There is an absolute constant $c>0$, such that the following statement is true. For any integer $d\ge 2$ and any real $\varepsilon$ satisfying $\frac{1}{4} \ge \varepsilon \ge \frac{1}{4\sqrt{d}}$, it holds that

Theorems & Definitions (8)

  • Theorem 1
  • Corollary 2
  • Remark
  • Theorem 3: AA
  • Claim 1
  • proof
  • Claim 2
  • proof