Title

On the Distribution of Integral Well-rounded Lattices in Dimension Two, Part I

Document Type

Lecture

Department

Mathematics (CMC)

Publication Date

9-27-2006

Abstract

A lattice is called well-rounded if its minimal vectors span the corresponding Eucildean space. Well-rounded lattices are very important objects in lattice theory in connection with packing and covering problems, as well as the famous conjecture of Minkowski, Frobenius problem, etc. In this talk we completely describe well-rounded full-rank sublattices of Z^2, as well as their determinant and minima sets. We show that the determinant set has positive density, deriving an explicit lower bound for it, while the minima set has density 0. We will also discuss formulas for the number of such lattices with a fixed determinant and with a fixed minimum. Our results extend automatically to well-rounded sublattices of any lattice AZ^2, where A is an element of the real orthogonal group O_2(R).

Comments

This lecture was given during the Number Theory Seminar at Texas A&M University in September 2006. It was continued in another lecture by the same author: "On the Distribution of Integral Well-rounded Lattices in Dimension Two, Part II" given during the Number Theory Seminar at Texas A&M University in February 2007.

Rights Information

© 2006 Lenny Fukshansky

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