Polynomial growth of sumsets in abelian semigroups
Number Theory
2016-12-30 v1 Combinatorics
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_1A_1 + >... + h_rA_r have multivariate polynomial growth.
Cite
@article{arxiv.math/0204052,
title = {Polynomial growth of sumsets in abelian semigroups},
author = {Melvyn B. Nathanson and Imre Z. Ruzsa},
journal= {arXiv preprint arXiv:math/0204052},
year = {2016}
}
Comments
8 pages. LaTex. To appear in Journal de Theorie des Nombres de Bordeaux