A note on discrete lattice-periodic sets with an application to Archimedean tilings

TL;DR

This paper explores the properties of discrete lattice-periodic sets, especially those arising from Archimedean tilings, extending classical results in the Geometry of Numbers to these structures.

Contribution

It generalizes Blichfeldt-type results to lattice-periodic sets derived from Archimedean tilings, providing new insights into their geometric and number-theoretic properties.

Findings

Archimedean tilings can be viewed as unions of lattice translates.

Established bounds for inhomogeneous problems in the Geometry of Numbers for these sets.

Illustrated the applicability of the results to classical tilings like the Archimedean types.

Abstract

Cao & Yuan obtained a Blichfeldt-type result for the vertex set of the edge-to-edge tiling of the plane by regular hexagons. Observing that every Archimedean tiling is the union of translates of a fixed lattice, we take a more general viewpoint and investigate basic questions for such point sets about the homogeneous and inhomogeneous problem in the Geometry of Numbers. The Archimedean tilings nicely exemplify our results.