On the area of empty axis-parallel rectangles amidst 2-dimensional lattice points

TL;DR

This paper studies the maximum empty rectangular areas amidst 2D lattice points, introducing a framework based on continued fractions to characterize and compute dispersion, with exact formulas and bounds for various lattices.

Contribution

It develops a comprehensive framework using continued fractions to analyze lattice dispersion, providing necessary and sufficient conditions, exact formulas, and bounds, and offers new insights into Zaremba's conjecture.

Findings

Characterization of finite dispersion lattices

Exact dispersion formulas for quadratic field subgroups

Bounds based on continued fraction coefficients

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 $n$-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.