A new bound for finite field Besicovitch sets in four dimensions

Terence Tao

Vol. 222 (2005), No. 2, 337–363

DOI: 10.2140/pjm.2005.222.337

Abstract

Let F be a finite field with characteristic greater than two. A Besicovitch set in F⁴ is a set P ⊆ F⁴ containing a line in every direction. The Kakeya conjecture asserts that P and F⁴ have roughly the same size, in the sense that |P|∕|F|⁴ exceeds C_𝜀|F|^−𝜀 for 𝜀 > 0 arbitrarily small, where C_𝜀 does not depend on P or F. Wolff showed that |P| exceeds a universal constant times |F|³. Here we improve his exponent to 3 + − 𝜀 for 𝜀 > 0 arbitrarily small. On the other hand, we show that Wolff’s 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 use of some basic algebraic geometry.

Keywords

Besicovitch sets, affine spaces over finite fields, Kakeya conjecture, reguli

Mathematical Subject Classification 2000

Primary: 42B25, 05C35

Milestones

Received: 19 April 2002

Revised: 10 September 2002

Published: 1 December 2005