On Distribution of Integral Well-Rounded Lattices in the Plane

Let Λ be a lattice of full rank in the N-dimensional Euclidean space RN for N ≥ 2. The minimum of Λ is defined as |Λ| := min{║x║ : x ∈ Λ \ {0}}, where ║║ stands for the usual Euclidean norm on RN, and the set of minimal vectors of Λ is defined to be S(Λ) := {x 2 Λ : ║x║ = |Λ|}. The lattice Λ is called well-rounded (abbreviated WR) if the set S(Λ) spans RN. WR lattices are important in discrete optimization, in particular in the investigation of sphere packing, sphere covering, and kissing number problems. Moreover, certain questions about distribution of WR lattices, along with the covering conjecture of Woods for WR lattices, came up recently in McMullen’s celebrated work on Minkowski’s conjecture. In this talk, we will discuss some results on distribution of integral WR lattices in the plane