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Let A and B be finite subsets of an algebraically closed field K of characteristic 0 such that |B| = C|A|. We show the following variant of the sum product phenomena: If |AB| < α|A| and α, C, α/C ≪ log |A|, then |kA + lB| ≫ |A| k |B| l. This is an application of a result of Evertse, Schlickewei, and Schmidt on linear equations with variables taking values… (More)

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