A new bound for finite field Besicovitch sets in four dimensions

Abstract

Let F be a finite field with characteristic greater than two. A Besicovitch set in F 4 is a set P ⊆ F 4 containing a line in every direction. The Kakeya conjecture asserts that P and F 4 have roughly the same size, in the sense that |P|/|F| 4 exceeds C e |F| -e for e > 0 arbitrarily small, where Cg 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+1 16 - e for e > 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.

Locations

• Pacific Journal of Mathematics - View
• arXiv (Cornell University) - PDF
• arXiv (Cornell University) - View
• Pacific Journal of Mathematics - View
• arXiv (Cornell University) - PDF
• arXiv (Cornell University) - View
• Pacific Journal of Mathematics - View
• arXiv (Cornell University) - PDF
• arXiv (Cornell University) - View