Homotheties and incidences

TL;DR

This paper investigates the combinatorial properties of rich homotheties in planar point sets, providing new bounds on their quantity and on the classification of k-element subsets under homothety transformations.

Contribution

It introduces a novel reduction of homothety problems to incidence problems in three-dimensional space, leading to refined bounds and extensions to higher dimensions.

Findings

New bounds for the number of rich homotheties

Bounds on the number of equivalence classes of k-element subsets

Extensions of results to higher dimensions

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 R^3, 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.