On Asymptotic Approximate Groups of Integers

2016/03/21 by Béla Bajnok, Bajnok, Bela
Mathematics · #Analytic Number Theory Research #FOS: Mathematics #Finite Group Theory Research #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1603.06553

openalex publication_date 2016/03/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let r be a positive integer, and let A be a nonempty finite set of at least two integers. We let C_r(A) denote the \em asymptotic r-covering number of A, that is, the smallest integer value of l for which, for all sufficiently large positive integers h, the rh-fold sumset of A is contained in at most l translates of the h-fold sumset of A. Nathanson proved that C_r(A) is always at most r+1; here we extend this result to prove that C_r(A) is always at least r, and determine all sets A for which C_r(A)=r.

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