Counting Lattice Points in Translations: Uniqueness and Existence Results

Source abstract

Generalizing a recent theorem by Higashitani et al. (Comb. Theory 2025), it is shown that a convex body $K$ is uniquely described up to integer translation by the function $\mathrm{t}_K(v) = |(K-v)\cap\mathbb{Z}^n|$, $v\in\mathbb{R}^n$. For a lattice polytope $P$ the function $\mathrm{t}_P$ is decomposed into homogeneous components $\mathrm{t}_{P,i}$, similar to the Ehrhart polynomial. It is shown that the functions $\mathrm{t}_{P,i}$ 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 $\mathrm{t}_{P,i}$. A structure theorem for the simultaneous eigenspace corresponding to the $(n-1)$st coefficient in the decomposition of $\mathrm{t}_P$ is obtained. This leads to an explicit description of the set of functions $\mathrm{t}_P$, where $P$ ranges over all lattice polygons, as well as to inequalities for the number of lattice points in dilated and translated lattice polytopes.