Quasi-polynomial counting of maximal lattice slices in higher dimensions

Establish that, for every lattice d-polytope P and every integer ℓ∈[d], the number of ℓ-dimensional slices of the dilation kP containing the maximum possible number of lattice points agrees, for all sufficiently large k, with a quasi-polynomial in k of degree d−ℓ.

Background

The paper proves the degree-one quasi-polynomial behavior of the number of lattice diameter lines in dilations of lattice polygons, corresponding to one-dimensional slices in dimension two. The authors conjecture a higher-dimensional and higher-codimension analogue for ℓ-dimensional slices, while noting that their proof techniques do not extend directly because maximal lattice slices need not pass through vertices in dimensions at least three and lattice-point counts are more subtly related to normalized volume.