On the number of rich lines in high dimensional real vector spaces

TL;DR

This paper uses polynomial partitioning to improve understanding of how many rich lines can exist in high-dimensional spaces, showing that many such lines must lie in a hyperplane if they are sufficiently numerous.

Contribution

It strengthens a recent result by demonstrating that a large number of rich lines in high dimensions are contained within a hyperplane, using polynomial partitioning techniques.

Findings

Many rich lines in high dimensions are contained in a hyperplane.

The number of rich lines is constrained by their incidence with points.

Polynomial partitioning effectively strengthens previous bounds.

Abstract

In this short note we use the polynomial partitioning lemma to strengthen a recent result of Dvir and Gopi about the number of rich lines in high dimensional Euclidean spaces. Our result shows that if there are sufficiently many rich lines incident to a set of points then large fraction of them must be contained in a hyperplane.