On Integral Well-rounded Lattices in the Plane

On distribution of integral well-rounded lattices in the plane

On Integral Well-rounded Lattices in the Plane

We investigate distribution of integral well-rounded lattices in the plane, producing a complete parameterization of the set of their similarity classes by solutions of the family of Pell-type Diophantine equations of the form x2 +Dy2 = z2 where D > 0 is squarefree. We then apply our results to the study of the greatest minimal norm and the highest signal-to-noise ratio on the set of such latti...

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 Distribution of Integral Well-Rounded Lattices in Dimension Two

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 ...