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

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 h1A1×⋯+hrAr en obtenant la croissance polynomiale du cardinal en termes des variables h1,h2,⋯,hr.

@article{JTNB_2002__14_2_553_0,
     author = {Melvyn B. Nathanson and Imre Z. Ruzsa},
     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},
     zbl = {1077.11014},
     mrnumber = {2040693},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/item/JTNB_2002__14_2_553_0/}
}
TY  - JOUR
AU  - Melvyn B. Nathanson
AU  - Imre Z. Ruzsa
TI  - Polynomial growth of sumsets in abelian semigroups
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2002
SP  - 553
EP  - 560
VL  - 14
IS  - 2
PB  - Université Bordeaux I
UR  - https://jtnb.centre-mersenne.org/item/JTNB_2002__14_2_553_0/
LA  - en
ID  - JTNB_2002__14_2_553_0
ER  - 
%0 Journal Article
%A Melvyn B. Nathanson
%A Imre Z. Ruzsa
%T Polynomial growth of sumsets in abelian semigroups
%J Journal de théorie des nombres de Bordeaux
%D 2002
%P 553-560
%V 14
%N 2
%I Université Bordeaux I
%U https://jtnb.centre-mersenne.org/item/JTNB_2002__14_2_553_0/
%G en
%F JTNB_2002__14_2_553_0
Melvyn B. Nathanson; Imre Z. Ruzsa. Polynomial growth of sumsets in abelian semigroups. Journal de théorie des nombres de Bordeaux, Tome 14 (2002) no. 2, pp. 553-560. https://jtnb.centre-mersenne.org/item/JTNB_2002__14_2_553_0/

[1] D. Cox, J. Little, D. O'Shea, Ideals, Varieties, and Algorithms. Springer-Verlag, New York, 2nd edition, 1997. | MR | Zbl

[2] S. Han, C. Kirfel, M.B. Nathanson, Linear forms in finite sets of integers. Ramanujan J. 2 (1998), 271-281. | MR | Zbl

[3] A.G. Khovanskii, Newton polyhedron, Hilbert polynomial, and sums of finite sets. Functional. Anal. Appl. 26 (1992), 276-281. | MR | Zbl

[4] A.G. Khovanskii, Sums of finite sets, orbits of commutative semigroups, and Hilbert functions. Functional. Anal. Appl. 29 (1995), 102-112. | MR | Zbl

[5] M.B. Nathanson, Sums of finite sets of integers. Amer. Math. Monthly 79 (1972), 1010-1012. | MR | Zbl

[6] M.B. Nathanson, Growth of sumsets in abelian semigroups. Semigroup Forum 61 (2000), 149-153. | MR | Zbl