no. 2
Polynomial growth of sumsets in abelian semigroups
Journal de théorie des nombres de Bordeaux, Tome 14 (2002) no. 2, pp. 553-560.

Soit S un semi-groupe abélien et A un sous-ensemble fini de S. On désigne par hA l’ensemble de toutes les sommes de h éléments de A, et par |hA| son cardinal. On montre, par des arguments élémentaires de comptage de points dans les réseaux, qu’il existe un polynôme p(t) tel que pour tout entier h assez grand |hA|=p(h). Plus généralement, on étend ce résultat aux ensembles h 1 A 1 ×⋯+h r A r en obtenant la croissance polynomiale du cardinal en termes des variables h 1 ,h 2 ,⋯,h r .

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 1 A 1 +⋯+h r A r have multivariate polynomial growth.

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     author = {Nathanson, Melvyn B. and Ruzsa, Imre Z.},
     title = {Polynomial growth of sumsets in abelian semigroups},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {553--560},
     publisher = {Universit\'e Bordeaux I},
     volume = {14},
     number = {2},
     year = {2002},
     mrnumber = {2040693},
     zbl = {1077.11014},
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     url = {http://www.numdam.org/item/JTNB_2002__14_2_553_0/}
}
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Nathanson, Melvyn B.; Ruzsa, Imre Z. Polynomial growth of sumsets in abelian semigroups. Journal de théorie des nombres de Bordeaux, Tome 14 (2002) no. 2, pp. 553-560. http://www.numdam.org/item/JTNB_2002__14_2_553_0/

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