Counting Lattice Points in Translations: Uniqueness and Existence Results

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 decom