Answering a problem posed by Keisler and Leth, we prove a theorem in non-standard analysis to reveal a phenomenon about sumsets, which says that if two sets A and B are large in terms of "measure", then the sum A+B is not small in terms of "order-topology". The theorem has several corollaries about sumset phenomenon in the standard world; these are described in sections 2-4. One of these is a new result in additive number theory; it says that if two sets A and B of non-negative integers have positive upper or upper Banach density, then A + B is piecewise syndetic. [References: 8]
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