Sets with few distinct distances do not have heavy lines

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 $O(n^{43/52}\mathrm{polylog}(n))$ 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.

References in corpus (4)

• On the Erdos distinct distance problem in the plane
• Distinct distances on algebraic curves in the plane
• On the Minkowski distances and products of sum sets
• On lattices, distinct distances, and the Elekes-Sharir framework

Cited by in corpus (3)

• New results on sum-products in R
• Bisector energy and few distinct distances
• An introduction to higher energies and sumsets