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

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

2007-08-12

arxiv.org/abs/0708.1596v2

Journal of Number Theory, vol. 129 no. 10 (2009), pg. 2530-2556

Mathematics

Number Theory

27 pages, 2 figures; added a lemma on Diophantine approximation by quotients of Pythagorean triples; final version to be publi

Scientific paper

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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