Homotheties and incidences

Abstract

We consider problems involving rich homotheties in a set S of n points in the plane (that is, homotheties that map many points of S to other points of S). By reducing these problems to incidence problems involving points and lines in R3, we are able to obtain refined and new bounds for the number of rich homotheties, and for the number of distinct equivalence classes, under homotheties, of k-element subsets of S, for any k≥3. We also discuss the extensions of these problems to three and higher dimensions.