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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 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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