On the Distribution of Integral Well-rounded Lattices in Dimension Two, Part I
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).
© 2006 Lenny Fukshansky
Fukshansky, Lenny. "On the Distribution of Integral Well-rounded Lattices in Dimension Two, Part I." Number Theory Seminar, Texas A&M University, College Station, Texas. 27 September 2006.