# Polynomial growth of sumsets in abelian semigroups

Melvyn B. Nathanson; Imre Z. Ruzsa

Journal de théorie des nombres de Bordeaux (2002)

- Volume: 14, Issue: 2, page 553-560
- ISSN: 1246-7405

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topNathanson, Melvyn B., and Ruzsa, Imre Z.. "Polynomial growth of sumsets in abelian semigroups." Journal de théorie des nombres de Bordeaux 14.2 (2002): 553-560. <http://eudml.org/doc/248890>.

@article{Nathanson2002,

abstract = {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 $h_1 A_1 + \cdots + h_r A_r$ have multivariate polynomial growth.},

author = {Nathanson, Melvyn B., Ruzsa, Imre Z.},

journal = {Journal de théorie des nombres de Bordeaux},

language = {eng},

number = {2},

pages = {553-560},

publisher = {Université Bordeaux I},

title = {Polynomial growth of sumsets in abelian semigroups},

url = {http://eudml.org/doc/248890},

volume = {14},

year = {2002},

}

TY - JOUR

AU - Nathanson, Melvyn B.

AU - Ruzsa, Imre Z.

TI - Polynomial growth of sumsets in abelian semigroups

JO - Journal de théorie des nombres de Bordeaux

PY - 2002

PB - Université Bordeaux I

VL - 14

IS - 2

SP - 553

EP - 560

AB - 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 $h_1 A_1 + \cdots + h_r A_r$ have multivariate polynomial growth.

LA - eng

UR - http://eudml.org/doc/248890

ER -

## References

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- [5] M.B. Nathanson, Sums of finite sets of integers. Amer. Math. Monthly79 (1972), 1010-1012. Zbl0251.10002MR304305
- [6] M.B. Nathanson, Growth of sumsets in abelian semigroups. Semigroup Forum61 (2000), 149-153. Zbl0959.20055MR1839220

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