A Note on Sequences of Lattices

Abstract

We investigate the relation between the convergence of a sequence of lattices and the set-theoretic convergence of their corresponding Voronoi cells sequence. We prove that if a sequence of full rank lattices converges to a full rank lattice, then the closures of the limit infimum and limit supremum of the Voronoi cells converges to the corresponding Voronoi cell. It remains an open question if the converse is also true.