Counting Lattice Points in Translations: Uniqueness and Existence Results

Abstract

Generalizing a recent theorem by Higashitani et al. (Comb. Theory 2025), it is shown that a convex body 𝛫 is uniquely described up to integer translation by the function tₖ(𝜈) = |(𝛫-𝜈)⋂ℤⁿ|, 𝜈∈ℝ. For a lattice polytope 𝛲 the function tₚ is decomposed into homogeneous components tₚ,ᵢ, similar to the Ehrhart polynomial. It is shown that the functions tₚ,ᵢ are simultaneous eigenfunctions of a family of linear operators introduced by Kubert and Lang in (Math. Z. 1976) and that, conversely, the linear spaces of piecewise constant eigenfunctions are generated by the respective coefficients tₚ,ᵢ. A structure theorem for the simultaneous eigenspace corresponding to the (𝑛-1)st coefficient in the decomposition of tₚ is obtained. This leads to an explicit description of the set of functions tₚ, where 𝛲 ranges over all lattice polygons, as well as to inequalities for the number of lattice points in dilated and translated lattice polytopes.