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In this note it is established that, for any finite set A of real numbers, there exist two elements a, b ∈ A such that |(a + A)(b + A)| |A| 2 log |A|. In particular, it follows that |(A + A)(A + A)| |A| 2 log |A|. The latter inequality had in fact already been established in an earlier work of the author and Rudnev [8], which built upon the recent… (More)

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