A finite version of the Kakeya problem

TL;DR

This paper constructs finite sets of lines with rich directional structure in affine spaces and establishes size bounds for point sets incident to these lines, extending the finite field Kakeya problem.

Contribution

It introduces a geometric construction of line sets with grid-like directions and provides size bounds for incident point sets based on polynomial ideals, generalizing finite Kakeya results.

Findings

Constructed line sets with N^{n-1} grid directions.

Derived lower bounds on point set sizes using polynomial ideals.

Extended finite Kakeya bounds to new configurations.

Abstract

Let $L$ be a set of lines of an affine space over a field and let $S$ be a set of points with the property that every line of $L$ is incident with at least $N$ points of $S$. Let $D$ be the set of directions of the lines of $L$ considered as points of the projective space at infinity. We give a geometric construction of a set of lines $L$, where $D$ contains an $N^{n-1}$ grid and where $S$ has size $2((1/2)N)^n$, given a starting configuration in the plane. We provide examples of such starting configurations for the reals and for finite fields. Following Dvir's proof of the finite field Kakeya conjecture and the idea of using multiplicities of Dvir, Kopparty, Saraf and Sudan, we prove a lower bound on the size of $S$ dependent on the ideal generated by the homogeneous polynomials vanishing on $D$. This bound is maximised as $((1/2)N)^n$ plus smaller order terms, for $n\geqslant 4$, when $D$ contains the points of a $N^{n-1}$ grid.