Well-rounded sublattices and coincidence site lattices

TL;DR

This paper explores the relationship between well-rounded sublattices and coincidence site lattices (CSLs), providing counts and asymptotic analysis for well-rounded sublattices in planar lattices.

Contribution

It establishes connections between well-rounded sublattices and CSLs and analyzes their enumeration and asymptotic properties in planar lattices.

Findings

Connections between well-rounded sublattices and CSLs established

Count of well-rounded sublattices provided for several planar lattices

Asymptotic behavior of well-rounded sublattice counts analyzed

Abstract

A lattice is called well-rounded, if its lattice vectors of minimal length span the ambient space. We show that there are interesting connections between the existence of well-rounded sublattices and coincidence site lattices (CSLs). Furthermore, we count the number of well-rounded sublattices for several planar lattices and give their asymptotic behaviour.