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

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 $p\left(t\right)$ such that $|hA|=p\left(h\right)$ for all sufficiently large $h$. Lattice point counting is also used to prove that sumsets of the form ${h}_{1}{A}_{1}+\cdots +{h}_{r}{A}_{r}$ have multivariate polynomial growth.

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 $p\left(t\right)$ tel que pour tout entier $h$ assez grand $|hA|=p\left(h\right)$. Plus généralement, on étend ce résultat aux ensembles ${h}_{1}{A}_{1}×\cdots +{h}_{r}{A}_{r}$ en obtenant la croissance polynomiale du cardinal en termes des variables ${h}_{1},{h}_{2},\cdots ,{h}_{r}$.

@article{JTNB_2002__14_2_553_0,
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},
publisher = {Universit\'e Bordeaux I},
volume = {14},
number = {2},
year = {2002},
pages = {553-560},
zbl = {1077.11014},
mrnumber = {2040693},
language = {en},
url = {http://www.numdam.org/item/JTNB_2002__14_2_553_0}
}

Nathanson, Melvyn B.; Ruzsa, Imre Z. Polynomial growth of sumsets in abelian semigroups. Journal de théorie des nombres de Bordeaux, Volume 14 (2002) no. 2, pp. 553-560. http://www.numdam.org/item/JTNB_2002__14_2_553_0/

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