A note on distinct distance subsets

TL;DR

This paper proves that for any set of N points in the plane or sphere, there exists a large subset where all pairwise distances are unique, with size roughly N^{1/3}/log N.

Contribution

It establishes a lower bound on the size of a subset with all distinct distances, advancing understanding of distance configurations in geometric point sets.

Findings

Existence of a subset of size ~N^{1/3}/log N with all distinct pairwise distances.

Applicable to points on the plane or sphere.

Provides a new lower bound for the problem of distinct distances in geometric sets.

Abstract

It is shown that given a set of $N$ points in the plane or on the sphere, there is a subset of size $\gtrsim N^{1/3}/\log N$ with all pairwise distances between points distinct.