Algorithm for counting generalized lattice vectors

Develop an algorithm that, given an n-dimensional lattice whose shortest nonzero vector has length 2 and a positive number α, determines the cardinality of the set X(α,Λ) = {v ∈ Λ : 2 ≤ ‖v‖ ≤ 2 + α}.

Background

The paper defines X(α,Λ) as the lattice vectors whose Euclidean lengths lie between 2 and 2 + α, for a lattice packing of the unit ball. The authors establish exact cardinalities of this set for several dimensions and specific lattices, but do not provide a general procedure for arbitrary lattices and positive α.

The proposed problem asks for an explicit algorithm capable of computing this finite lattice-point count under the normalization that the shortest nonzero lattice vector has length 2. It is motivated by the observation that, for almost every lattice, the kissing number is only 2, whereas the number of vectors in a larger annulus can be substantially different.