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 K is uniquely described up to integer translation by the function tK(v) = |(K−v)∩ℤn|, v∈ℝn. For a lattice polytope P 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 P ranges over all lattice polygons, as well as to inequalities for the number of lattice points in dilated and translated lattice polytopes.
Related
- Existence and uniqueness for non-linear forward equations: Addendum to "Mean-field approximations in insurance" (Hornung, 2026)
- Long-time existence and uniqueness of the heterotic G2 flow
- A quasi-linear Schrödinger equation for nucleons : existence, uniqueness and asymptotic behavior
- Existence and uniqueness of solutions to Hardy-Henon boundary-value problems
- Bi-invariant Geodesic Regression: Existence, Uniqueness, and Convergence
BibTeX
@misc{freyer2026counting,
title = {{Counting Lattice Points in Translations: Uniqueness and Existence Results}},
author = {Ansgar Freyer and Georg C. Hofstätter},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26523},
eprint = {2610.26523},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26523},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Counting Lattice Points in Translations: Uniqueness and Existence Results
AU - Ansgar Freyer
AU - Georg C. Hofstätter
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26523
UR - https://arxiv.org/abs/2610.26523
AB - 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 t_K(v) = |(K−v)∩ℤ^n|, v∈ℝ^n. For a lattice polytope P the function t_P is decomposed into homogeneous components t_P,i, similar to the Ehrhart polynomial. It is shown that the functions 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 t_P,i. A structure theorem for the simultaneous eigenspace corresponding to the (n−1)st coefficient in the decomposition of t_P is obtained. This leads to an explicit description of the set of functions 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.
ER -