Collinear Triple Hypergraphs and the Finite Plane Kakeya Problem
组合数学
2007-05-23 v3 数论
摘要
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.
引用
@article{arxiv.math/0607734,
title = {Collinear Triple Hypergraphs and the Finite Plane Kakeya Problem},
author = {Joshua N. Cooper},
journal= {arXiv preprint arXiv:math/0607734},
year = {2007}
}
备注
17 pages, no figures. Typos fixed, arithmetic errors fixed. A big thanks to Xander Faber for his help