On the area of empty axis-parallel rectangles amidst 2-dimensional lattice points
Abstract
The dispersion of a point set in the unit square is defined to be the area of the largest empty axis-parallel box. In this paper we are interested in the dispersion of lattices in the plane, that is, the supremum of the area of the empty axis-parallel boxes amidst the lattice points. We introduce a framework with which to study this based on the continued fractions expansions of the generators of the lattice. This framework proves so successful that we were unable to ask a question that we could not answer. We give necessary and sufficient conditions under which a lattice has finite dispersion. We obtain an exact formula for the dispersion of the lattices associated to subgroups of the ring of integer of a quadratic field. We have tight bounds for the dispersion of a lattice based the largest continued fraction coefficient of the generators, accurate to within one half. We know what the -th best lattice is. We provide an equivalent formulation of Zaremba's conjecture. Using our framework we are able to give alternative proofs of the results from two other papers in only a few lines.
Keywords
Cite
@article{arxiv.2109.11222,
title = {On the area of empty axis-parallel rectangles amidst 2-dimensional lattice points},
author = {Thomas Lachmann and Jaspar Wiart},
journal= {arXiv preprint arXiv:2109.11222},
year = {2021}
}