2002/04/19 by Terence Tao, Tao, Terence · 2 citations
Computer Science · Engineering · Mathematics · #05C35 #42B25 #Classical Analysis and ODEs (math.CA) #Digital Image Processing Techniques #FOS: Mathematics #Limits and Structures in Graph Theory #graph theory and CDMA systems #math.CA #msc:05C35 #msc:42B25
paper · pdf · doi:10.48550/arxiv.math/0204251
28 pages, no figures, to appear, Pacific J. Math. More exposition, less typos
openalex publication_date 2002/04/19 · arxiv created 2002/09/10 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let F be a finite field with characteristic greater than two. Define a Besicovitch set in F4 to be a set P ⊆ F4 containing a line in every direction. The Kakeya conjecture asserts that |P| ≈ |F|4. A result of Wolff establishes that |P| \gtrsim |F|3. In this paper we improve this to |P| \gtrapprox |F|3+\gain. On the other hand, we show that the bound of |F|3 is sharp if we relax the assumption that the lines point in different directions. One new feature in the argument is the introduction of a small amount of basic algebraic geometry.