On similarity classes of well-rounded sublattices of $\mathbb Z^2$

TL;DR

This paper explores the structure and distribution of well-rounded sublattices of Z^2, relating them to Pythagorean triples, and introduces a zeta function to analyze their growth and packing density.

Contribution

It establishes a connection between similarity classes of well-rounded sublattices and primitive Pythagorean triples, revealing a noncommutative monoid structure and defining related zeta functions.

Findings

Similarity classes form a noncommutative infinitely generated monoid.

A zeta function related to Dedekind zeta of Z[i] is introduced.

Constructed sequences of lattices approach optimal circle packing density.

Abstract

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 ${\mathbb Z}^2$. We relate the set of all such similarity classes to a subset of primitive Pythagorean triples, and prove that it has 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 ${\mathbb Z}[i]$, and investigate the growth of some related Dirichlet series, which reflect on the distribution of well-rounded lattices. Finally, we construct a sequence of similarity classes of well-rounded sublattices of ${\mathbb Z}^2$, which gives good circle packing density and converges to the hexagonal lattice as fast as possible with respect to a natural metric we define.