Abstract
Let A and B be two finite sets of numbers. The sum set and the product set of A, B are A + B = {a + b : a in A, b in B}, and AB = {ab : a in A, b in B}. $ We prove that A+B is as large as possible when AA is not too big. Similarly, AB is large when A+A is not too big. The methods rely on the Lambda_p constant of A, bound on the number of factorizations in a generalized progression containing A, and the subspace theorem.
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