Sets with few distinct distances do not have heavy lines

TL;DR

This paper proves that point sets with very few distinct distances cannot have lines containing too many points, extending to rectangular distances in Minkowski geometry, revealing structural limitations of such configurations.

Contribution

It establishes upper bounds on the number of points from a low-distance set that can lie on a single line, including for rectangular distances in Minkowski space.

Findings

No line contains more than O(n^{43/52} polylog(n)) points from the set.

Similar bounds hold for rectangular distances in Minkowski space.

Sets with few distances have restricted collinearity.

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 $O(n^{43/52}{\rm polylog}(n))$ points of $P$. We also show a similar result for rectangular distances, equivalent to distances in the Minkowski plane, where the distance between a pair of points is the area of the axis-parallel rectangle that they span.