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Polynomial growth of sumsets in abelian semigroups

Melvyn B. Nathanson·Imre Z. Ruzsa
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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 h_1A_1 + >... + h_rA_r have multivariate polynomial growth.

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Melvyn B. Nathanson, Imre Z. Ruzsa. 2002-04-03. Polynomial growth of sumsets in abelian semigroups. https://arxiv.org/abs/math/0204052

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