Generalizations of Khovanskiĭ's theorems on the growth of sumsets in Abelian semigroups

We show that if P is a lattice polytope in the nonnegative orthant of Rk and χ is a coloring of the lattice points in the orthant such that the color χ(a+b) depends only on the colors χ(a) and χ(b), then the number of colors of the lattice points in the dilation nP of P is for large n given by a polynomial (or, for rational P, by a quasipolynomial). This unifies a classical result of Ehrhart and Macdonald on lattice points in polytopes and a result of Khovanskiĭ on sumsets in semigroups. We also prove a strengthening of multivariate generalizations of Khovanskiĭ's theorem. Another result of Khovanskiĭ states that the size of the image of a finite set after n applications of mappings from a finite family of mutually commuting mappings is for large n a polynomial. We give a combinatorial proof of a multivariate generalization of this theorem.