The sumset phenomenon (Q2759038)

Using Robinson-Zakon styled nonstandard analysis, the author establishes a single major theorem in terms of the usual language used within this style of nonstandard analysis. However, this result establishes various standard sumset styled results. Using this theorem, the author establishes the known result that if subsets \(A,\) \(B\) of the real numbers have positive Lebesgue measure, then there is a nonempty open interval contained in \(A + B.\) The major theorem is then applied to obtain new results relative to additions on a sequence of cyclic groups and in additive number theory. In particular, it is shown that if \(A\) and \(B\) are subsets of the natural numbers and both have positive upper Banach density, then \(A+B\) is piecewise syndetic.