On Similarity Classes of Well-rounded Sublattices of Z

A lattice is called well-rounded if its minimal vectors span the corresponding Euclidean space. In this paper we study the similarity classes of well-rounded sublattices of Z2. We relate the set of all such similarity classes to a subset of primitive Pythagorean triples, and prove that it has the structure of a noncommutative infinitely generated monoid. We discuss the structure of a given similarity class, and define a zeta function corresponding to each similarity class. We relate it to Dedekind zeta of Z[i], and investigate the growth of some related Dirichlet series, which reflect on the distribution of well-rounded lattices. We also construct a sequence of similarity classes of well-rounded sublattices of Z2, which gives good circle packing density and converges to the hexagonal lattice as fast as possible with respect to a natural metric we define. Finally, we discuss distribution of similarity classes of wellrounded sublattices of Z2 in the set of similarity classes of all well-rounded lattices in R2.