On Asymptotic Approximate Groups of Integers
Bela Bajnok

TL;DR
This paper studies the asymptotic r-covering number of finite sets of integers, establishing bounds and characterizing sets where this number equals r, thus extending Nathanson's earlier results.
Contribution
It extends Nathanson's result by proving the lower bound of the asymptotic r-covering number and characterizes sets where this number equals r.
Findings
The asymptotic r-covering number is always at least r.
It is always at most r+1, as previously shown.
Sets with asymptotic r-covering number exactly r are fully characterized.
Abstract
Let be a positive integer, and let be a nonempty finite set of at least two integers. We let denote the {\em asymptotic -covering number} of , that is, the smallest integer value of for which, for all sufficiently large positive integers , the -fold sumset of is contained in at most translates of the -fold sumset of . Nathanson proved that is always at most ; here we extend this result to prove that is always at least , and determine all sets for which .
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Taxonomy
TopicsLimits and Structures in Graph Theory · Analytic Number Theory Research · Finite Group Theory Research
