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arXiv · 1410.1654

Sets with few distinct distances do not have heavy lines

Orit E. Raz·Oliver Roche-Newton·Micha Sharir
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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.

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Orit E. Raz, Oliver Roche-Newton, Micha Sharir. 2014-10-07. Sets with few distinct distances do not have heavy lines. https://arxiv.org/abs/1410.1654

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