Let be an abelian semigroup, and a finite subset of . The sumset consists of all sums of elements of , with repetitions allowed. Let denote the cardinality of . Elementary lattice point arguments are used to prove that an arbitrary abelian semigroup has polynomial growth, that is, there exists a polynomial such that for all sufficiently large . Lattice point counting is also used to prove that sumsets of the form have multivariate polynomial growth.
Soit un semi-groupe abélien et un sous-ensemble fini de . On désigne par l’ensemble de toutes les sommes de éléments de , et par son cardinal. On montre, par des arguments élémentaires de comptage de points dans les réseaux, qu’il existe un polynôme tel que pour tout entier assez grand . Plus généralement, on étend ce résultat aux ensembles en obtenant la croissance polynomiale du cardinal en termes des variables .
@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, Volume 14 (2002) no. 2, pp. 553-560. https://jtnb.centre-mersenne.org/item/JTNB_2002__14_2_553_0/
[1] Ideals, Varieties, and Algorithms. Springer-Verlag, New York, 2nd edition, 1997. | MR | Zbl
, , ,[2] Linear forms in finite sets of integers. Ramanujan J. 2 (1998), 271-281. | MR | Zbl
, , ,[3] Newton polyhedron, Hilbert polynomial, and sums of finite sets. Functional. Anal. Appl. 26 (1992), 276-281. | MR | Zbl
,[4] Sums of finite sets, orbits of commutative semigroups, and Hilbert functions. Functional. Anal. Appl. 29 (1995), 102-112. | MR | Zbl
,[5] Sums of finite sets of integers. Amer. Math. Monthly 79 (1972), 1010-1012. | MR | Zbl
,[6] Growth of sumsets in abelian semigroups. Semigroup Forum 61 (2000), 149-153. | MR | Zbl
,