Quasi-polynomial counting of maximal lattice-point slices

Prove that for every lattice d-polytope P and every integer 1≤ℓ≤d, the number of ℓ-dimensional slices of kP containing the maximum number of lattice points among all ℓ-dimensional slices agrees, for all sufficiently large k, with a quasi-polynomial in k of degree d−ℓ.

Background

The paper establishes an Ehrhart-type quasi-polynomial result for counting lattice diameter lines, which are one-dimensional maximal lattice-point slices, in dilations of lattice polygons. It proposes extending this behavior to arbitrary slice dimensions and ambient dimensions. The authors explicitly identify technical obstacles in higher dimensions, including the possible absence of diameter segments through vertices and the more complicated relationship between normalized volume and lattice-point counts.

References

We conjecture that counting lattice diameter segments or more generally counting $\ell$-slices of $P$ that contain the most lattice points among all $\ell$-dimensional slices of $P$ have a similar behavior under dilation of $P$.

— On Lattice Diameter Segments and A Discrete Borsuk Partition Problem  (2508.20009 - Brose et al., 27 Aug 2025) in Section 3, immediately preceding the second conjecture