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Suppose is a set of non-negative integers with upper Banach density (see definition below) and the upper Banach density of is less than . We characterize the structure of by showing the following: There is a positive integer and a set , which is the union of arithmetic sequences [We call a set of the form an arithmetic sequence of difference and call a set of the form an arithmetic progression of difference . So an arithmetic progression is finite and an arithmetic sequence...
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