On sets defining few ordinary planes

TL;DR

This paper characterizes the structure of large point sets in three-dimensional space with few ordinary planes, showing they are mostly contained in intersections of quadrics or specific prism-like configurations, and relates this to planar sets with few triple-circle incidences.

Contribution

It establishes structural theorems for point sets with few ordinary planes and extends these ideas to planar sets with few triple-circle incidences, identifying specific geometric configurations.

Findings

Most points lie in the intersection of two quadrics when few ordinary planes are present.

Point sets with very few triple incidences are either prisms, anti-prisms, or similar structures.

In the plane, sets with few triple-circle incidences are mostly on low-degree algebraic curves.

Abstract

Let $S$ be a set of $n$ points in real three-dimensional space, no three collinear and not all co-planar. We prove that if the number of planes incident with exactly three points of $S$ is less than $Kn^2$ for some $K=o(n^{\frac{1}{7}})$ then, for $n$ sufficiently large, all but at most $O(K)$ points of $S$ are contained in the intersection of two quadrics. Furthermore, we prove that there is a constant $c$ such that if the number of planes incident with exactly three points of $S$ is less than $\frac{1}{2}n^2-cn$ then, for $n$ sufficiently large, $S$ is either a prism, an anti-prism, a prism with a point removed or an anti-prism with a point removed. As a corollary to the main result, we deduce the following theorem. Let $S$ be a set of $n$ points in the real plane. If the number of circles incident with exactly three points of $S$ is less than $Kn^2$ for some $K=o(n^{\frac{1}{7}})$ then, for $n$ sufficiently large, all but at most $O(K)$ points of $S$ are contained in a curve of degree at most four.