Improved dispersion bounds for modified Fibonacci lattices

TL;DR

This paper improves the known bounds on the dispersion of point sets in the unit square by modifying Fibonacci lattices, achieving a lower dispersion than previously known, which advances understanding of optimal point distributions.

Contribution

The authors introduce a modification of Fibonacci lattices that significantly reduces dispersion, improving the upper bound on the minimal dispersion for large point sets.

Findings

Modified Fibonacci lattices have lower dispersion than standard ones.

New upper bound for dispersion limit is approximately 1.8944.

Results suggest better point set constructions for uniform sampling.

Abstract

We study the dispersion of point sets in the unit square; i.e. the size of the largest axes-parallel box amidst such point sets. It is known that $\liminf_{N\to\infty} N\mathrm{disp}(N,2)\in \left[\frac54,2\right],$ where $\mathrm{disp}(N,2)$ is the minimal possible dispersion for an $N$-element point set in the unit square. The upper bound 2 is obtained by an explicit point construction - the well-known Fibonacci lattice. In this paper we find a modification of this point set such that its dispersion is significantly lower than the dispersion of the Fibonacci lattice. Our main result will imply that $\liminf_{N\to\infty} N\mathrm{disp}(N,2)\leq \varphi^3/\sqrt{5}=1.894427...$