Sets with few distinct distances do not have heavy lines
Orit E. Raz, Oliver Roche‐Newton, Micha Sharir · arXiv (Cornell University) · 2014
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.