Kneser’s Theorem For Upper Banach Density

Abstract

Suppose A is a set of non-negative integers with upper Banach density α (see definition below) and the upper Banach density of A + A is less than 2α. We characterize the structure of A+A by showing the following: There is a positive integer g and a set W , which is the union of d2αg − 1e arithmetic sequences1 with the same difference g such that A + A ⊆ W and if [an, bn] for each n is an interval of integers such that bn − an → ∞ and the relative density of A in [an, bn] approaches α, then there is an interval [cn, bn] ⊆ [an, bn] for each n such that (dn − cn)/(bn − an) → 1 and (A+ A)∩ [2cn, 2dn] = W ∩ [2cn, 2dn].

Cite this paper

@inproceedings{Bihani2006KnesersTF, title={Kneser’s Theorem For Upper Banach Density}, author={Prerna Bihani and Renling Jin}, year={2006} }