Oliver Roche-Newton

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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)
A variation on the sum-product problem seeks to show that a set which is defined by additive and multiplicative operations will always be large. In this paper, we prove new results of this type. In particular, we show that for any finite set A of positive real numbers, it is true that ∣ ∣ ∣ a + b c + d : a, b, c, d ∈ A }∣ ∣ ∣ ≥ 2|A|2 − 1. As a consequence(More)