Polynomial Wolff axioms and Kakeya-type estimates in R4

Abstract

We establish new linear and trilinear bounds for collections of tubes in R 4 that satisfy the polynomial Wolff axioms. In brief, a collection of δ-tubes satisfies the Wolff axioms if not too many tubes can be contained in the δ-neighborhood of a plane. A collection of tubes satisfies the polynomial Wolff axioms if not too many tubes can be contained in the δ-neighborhood of a low degree algebraic variety. First, we prove that if a set of δ − 3 tubes in R 4 satisfies the polynomial Wolff axioms, then the union of the tubes must have volume at least δ 1 − 1 / 28 . We also prove a more technical statement which is analogous to a maximal function estimate at dimension 3 + 1 / 28 . Second, we prove that if a collection of δ − 3 tubes in R 4 satisfies the polynomial Wolff axioms, and if most triples of intersecting tubes point in three linearly independent directions, then the union of the tubes must have volume at least δ 3 / 4 . Again, we also prove a slightly more technical statement which is analogous to a maximal function estimate at dimension 3 + 1 / 4 . We conjecture that every Kakeya set satisfies the polynomial Wolff axioms, but we are unable to prove this. If our conjecture is correct, it implies a Kakeya maximal function estimate at dimension 3 + 1 / 28 , and in particular this implies that every Kakeya set in R 4 must have Hausdorff dimension at least 3 + 1 / 28 . This would be an improvement over the current best bound of 3, which was established by Wolff in 1995.

Locations

• Proceedings of the London Mathematical Society - View
• arXiv (Cornell University) - View - PDF
• DSpace@MIT (Massachusetts Institute of Technology) - View - PDF
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