Every finite subset of an abelian group is an asymptotic approximate group

JOURNAL OF NUMBER THEORY

卷:191

Article

Nathanson, Melvyn B.¹ 

[1] CUNY, Lehman Coll, Dept Math, Bronx, NY 10468 USA

关键词: Approximate group;    Additive number theory;    Sumsets;    Lattice polytope;    Abelian group;   

DOI  :  10.1016/j.jnt.2018.04.014

来源: Elsevier

【 摘 要 】

If A is a nonempty subset of an additive abelian group G, then the h-fold sumset is hA = {x(1) + ... + x(h) : x(i) is an element of A(i) for i = 1, 2,.. ., h}. We do not assume that A contains the identity, nor that A is symmetric, nor that A is finite. The set A is an (r,l)-approximate group in G if there exists a subset X of G such that |X| <= l and rA subset of XA. The set A is an asymptotic (r,l)-approximate group if the sumset hA is an (r,l)-approximate group for all sufficiently large h. It is proved that every polytope in a real vector space is an asymptotic (r,l)-approximate group, that every finite set of lattice points is an asymptotic (r,l)-approximate group, and that every finite subset of every abelian group is an asymptotic (r,l)-approximate group. (C) 2018 Published by Elsevier Inc.

【 授权许可】

Free