Improved dispersion bounds for modified Fibonacci lattices

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 lim infN→∞Ndisp(N,2)∈54,2, where 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 lim infN→∞Ndisp(N,2)≤φ3∕5=1.894427...