Collinear Triple Hypergraphs and the Finite Plane Kakeya Problem

Details Collinear Triple Hypergraphs and the Finite Plane Kakeya Problem Collinear Triple Hypergraphs and the Finite Plane Kakeya Problem

2006-07-28

arxiv.org/abs/math/0607734v3

Mathematics

Combinatorics

17 pages, no figures. Typos fixed, arithmetic errors fixed. A big thanks to Xander Faber for his help

Scientific paper

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.

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