On Cartesian Products which Determine Few Distinct Distances

2016/12/19 by Pohoata, Cosmin
#Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1612.06153

Abstract

Every set of points P determines Ω(|P| / log |P|) distances. A close version of this was initially conjectured by Erdős in 1946 and rather recently proved by Guth and Katz. We show that when near this lower bound, a point set P of the form A × A must satisfy |A - A| ≪ |A|^2-(2)/(7) log^(1)/(7) |A|. This improves recent results of Hanson and Roche-Newton.

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