Cyclic and well-rounded lattices

On Well-rounded Ideal Lattices

We investigate a connection between two important classes of Euclidean lattices: well-rounded and ideal lattices. A lattice of full rank in a Euclidean space is called well-rounded if its set of minimal vectors spans the whole space. We consider lattices coming from full rings of integers in number fields, proving that only cyclotomic fields give rise to well-rounded lattices. We further study ...

Well - Rounded Integral Lattices in Dimension Two

A lattice is called well-rounded if its minimal vectors span the corresponding Eucildean space. In this paper 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 also produce formulas for the number of...

On Well-rounded Ideal Lattices - Ii

We study well-rounded lattices which come from ideals in quadratic number fields, generalizing some recent results of the first author with K. Petersen [8]. In particular, we give a characterization of ideal well-rounded lattices in the plane and show that a positive proportion of real and imaginary quadratic number fields contains ideals giving rise to well-rounded lattices.

Closed Orbits for the Diagonal Group and Well-rounded Lattices

Curt McMullen showed that every compact orbit for the action of the diagonal group on the space of lattices contains a well-rounded lattice. We extend this to all closed orbits.

Minkowski’s Conjecture, Well-rounded Lattices and Topological Dimension