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Kneser’s theorem for upper Banach density

Prerna Bihani; Renling Jin — 2006

Journal de Théorie des Nombres de Bordeaux

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 $2 α$ . 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 $⌈ 2 α g - 1 ⌉$ arithmetic sequences [We call a set of the form $a + d ℕ$ an arithmetic sequence of difference $d$ and call a set of the form ${a, a + d, a + 2 d, ..., a + k d}$ an arithmetic progression of difference $d$ . So an arithmetic progression is finite and an arithmetic sequence...

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