#### Title

On Similarity Classes of Well-Rounded Sublattices of Z²

#### Document Type

Article

#### Department

Mathematics (CMC)

#### Publication Date

10-2009

#### 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 *Z*^{2}. 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 non-commutative 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 *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. Finally, we discuss distribution of similarity classes of well-rounded sublattices of *Z*^{2} in the set of similarity classes of all well-rounded lattices in *R*^{2}.

#### Rights Information

© 2009 Elsevier Inc.

#### Terms of Use & License Information

#### Recommended Citation

Fukshansky, Lenny. "On Similarity Classes of Well-Rounded Sublattices of Z²." Journal of Number Theory 129.10 (2009): 2530-2556.

## Comments

More information on this article can be found at http://www.sciencedirect.com/science/article/pii/S0022314X09000730

This article relates to a lecture ("On similarity classes of well-rounded sublattices of Z

^{2}") given by Lenny Fukshansky during the Oberseminar des Institutes für Algebra und Geometrie at the University of Magdeburg, Germany in June 2008.