Collinear Triple Hypergraphs and the Finite Plane Kakeya Problem

TL;DR

This paper links the finite plane Kakeya problem with collinear point counts in permutations, providing new lower bounds for collinear triples and Besicovitch sets in finite fields using hypergraph analysis.

Contribution

It introduces a novel connection between collinear triples in permutations and the Kakeya problem, deriving improved lower bounds through hypergraph counting methods.

Findings

New lower bound (5q/14 + O(1)) for collinear triples in permutations.

New lower bound (q(q + 1)/2 + 5q/14 + O(1)) for smallest Besicovitch sets.

Establishes structural questions about collinear triple hypergraphs.

Abstract

We show that the problem of counting collinear points in a permutation (previously considered by the author and J. Solymosi in "Collinear Points in Permutations", 2005) and the well-known finite plane Kakeya problem are intimately connected. Via counting arguments and by studying the hypergraph of collinear triples we show a new lower bound (5q/14 + O(1)) for the number of collinear triples of a permutation of GF(q) and a new lower bound (q(q + 1)/2 + 5q/14 + O(1)) on the size of the smallest Besicovitch set in GF(q)^2. Several interesting questions about the structure of the collinear triple hypergraph are presented.