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

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.