# Polynomial growth of sumsets in abelian semigroups

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

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## Abstract

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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.

## How to cite

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Nathanson, 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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1. [1] D. Cox, J. Little, D. O'Shea, Ideals, Varieties, and Algorithms. Springer-Verlag, New York, 2nd edition, 1997. Zbl0861.13012MR1417938
2. [2] S. Han, C. Kirfel, M.B. Nathanson, Linear forms in finite sets of integers. Ramanujan J.2 (1998), 271-281. Zbl0911.11008MR1642882
3. [3] A.G. Khovanskii, Newton polyhedron, Hilbert polynomial, and sums of finite sets. Functional. Anal. Appl.26 (1992), 276-281. Zbl0809.13012MR1209944
4. [4] A.G. Khovanskii, Sums of finite sets, orbits of commutative semigroups, and Hilbert functions. Functional. Anal. Appl.29 (1995), 102-112. Zbl0855.13011MR1340302
5. [5] M.B. Nathanson, Sums of finite sets of integers. Amer. Math. Monthly79 (1972), 1010-1012. Zbl0251.10002MR304305
6. [6] M.B. Nathanson, Growth of sumsets in abelian semigroups. Semigroup Forum61 (2000), 149-153. Zbl0959.20055MR1839220

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