Polynomial growth of sumsets in abelian semigroups
Melvyn B. Nathanson, Imre Z. Ruzsa
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.
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