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 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 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 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 well-rounded sublattices of Z2 in the set of similarity classes of all well-rounded lattices in R2.
Rights Information
© 2009 Elsevier Inc.
Terms of Use & License Information
DOI
10.1016/j.jnt.2009.01.023
Recommended Citation
Fukshansky, Lenny. "On Similarity Classes of Well-Rounded Sublattices of Z²." Journal of Number Theory 129.10 (2009): 2530-2556. https://doi.org/10.1016/j.jnt.2009.01.023
Comments
More information on this article can be found at https://www.sciencedirect.com/science/article/pii/S0022314X09000730
This article relates to a lecture ("On similarity classes of well-rounded sublattices of Z2") given by Lenny Fukshansky during the Oberseminar des Institutes für Algebra und Geometrie at the University of Magdeburg, Germany in June 2008.