Polynomial growth of sumsets in abelian semigroups

Abstract

Let S be an abelian semigroup, and A a finite subset of S. The sumset hA consists of all sums of h elements of A, with repetitions allowed. Let |hA| denote the cardinality of hA. Elementary lattice point arguments are used to prove that an arbitrary abelian semigroup has polynomial growth, that is, there exists a polynomial p(t) such that |hA| = p(h) for all sufficiently large h. Lattice point counting is also used to prove that sumsets of the form h1A1 + • • • + hrAr have multivariate polynomial growth.

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

“…We build on the results of Khovanskiȋ [9,10], Nathanson and Ruzsa [17] and Stanley [19]. Khovanskiȋ's original proof of part 2 of Theorem 1.2 as a corollary of Theorem 1.4 in [9] was algebraic, by means of the Hilbert polynomial of graded modules.…”

Additive combinatorics and geometry of numbers

“…This means that the hull volume can be defined without any reference to convexity and measure, and this definition can even be extended to commutative semigroups. This follows from the following result of Khovanskii [5], [6]; for a simple proof see [8].…”

John-type theorems for generalized arithmetic progressions and iterated sumsets

“…It may be that one of these exponential losses can be removed, or that two of them can be run "in parallel", reducing the total loss to a double exponential, but we will not attempt to do so here. In the asymptotic limit l → ∞, much more about the structure of lA is known, for instance |lA| is eventually a polynomial in l [12], [13]. The behavior of lA for large l is also closely connected to Theorems 1.15, 1.16; see [9] for further discussion.…”