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arXiv · 0708.1596

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

Lenny Fukshansky
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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.

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Lenny Fukshansky. 2009-02-09. On similarity classes of well-rounded sublattices of $\mathbb Z^2$. https://arxiv.org/abs/0708.1596

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