# A Note on Sequences of Lattices

## TL;DR

This paper explores how the convergence of lattice sequences relates to the set-theoretic convergence of their Voronoi cells, establishing a one-way implication and highlighting an open question about the converse.

## Contribution

It proves that lattice convergence implies Voronoi cell convergence, and identifies an open problem regarding the converse relationship.

## Key findings

- Lattice convergence implies Voronoi cell convergence.
- The closure of limit infimum and supremum of Voronoi cells converge to the Voronoi cell.
- Open question remains about the converse implication.

## 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.