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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 , which built upon the recent… (More)
Sets with few distinct distances do not have heavy lines Abstract Let P be a set of n points in the plane that determines at most n/5 distinct distances. We show that no line can contain more than O(n 43/52 polylog(n)) points of P. We also show a similar result for rectangular distances, equivalent to distances in the Minkowski plane, where the distance… (More)
We prove several expanders with exponent strictly greater than 2. For any finite set A ⊂ R, we prove the following six-variable expander results: |(A−A)(A−A)(A−A)| |A| 2+ 1 8 log 17 16 |A| , ∣∣∣∣A+A A+A + AA ∣∣∣∣ |A| 2 17 log 16 17 |A| , ∣∣∣∣AA+AA A+A ∣∣∣∣ |A| 18 log |A| , ∣∣∣∣AA+A AA+A ∣∣∣∣ |A| 18 log |A| .