2006/07/28 by Joshua N. Cooper, Cooper, Joshua N.
Mathematics · #14N10 #51E15 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #math.CO #math.NT #msc:14N10 #msc:51E15
paper · pdf · doi:10.48550/arxiv.math/0607734
17 pages, no figures. Typos fixed, arithmetic errors fixed. A big thanks to Xander Faber for his help
arxiv created 2006/08/14 · arxiv updated 2009/12/01
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.