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Polynomial growth of sumsets in abelian semigroups

Melvyn B. Nathanson, Imre Z. Ruzsa (2002)

Journal de théorie des nombres de Bordeaux

Let S be an abelian semigroup, and A a finite subset of S . The sumset h A consists of all sums of h elements of A , with repetitions allowed. Let | h A | denote the cardinality of h A . 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 | h A | = p ( h ) for all sufficiently large h . Lattice point counting is also used to prove that sumsets of the form h 1 A 1 + + h r A r have multivariate polynomial growth.

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