Growth of sumsets in abelian semigroups

Abstract

Let S be an abelian semigroup, written additively. Let A be a finite subset of S. We denote the cardinality of A by |A|. For any positive integer h, the sumset hA is the set of all sums of h not necessarily distinct elements of A. We define 0A = {0}. If A_1,...,A_r, and B are finite sumsets of A and h_1,...,h_r are nonnegative integers, the sumset h_1A + ... + h_rA_r + B is the set of all elements of S that can be represented in the form u_1 + ... + u_r + b, where u_i \in h_iA_i and b \in B. The growth function of this sumset is γ(h_1,...,h_r) = |h_1A + ... + h_rA_r + B|. Applying the Hilbert function for graded modules over graded algebras, where the grading is over the semigroup of r-tuples of nonnegative integers, we prove that there is a polynomial p(t_1,...,t_r) such that γ(h_1,...,h_r) = p(t_1,...,t_r) if min(h_1,...,h_r) is sufficienlty large.