Polynomial Wolff axioms and Kakeya-type estimates in ${\mathbb{R}}^4$

Abstract

We establish new linear and trilinear bounds for collections of tubes in ${\mathbb{R}}^4$ that satisfy the polynomial Wolff axioms. In brief, a collection of $\delta$-tubes satisfies the Wolff axioms if not too many tubes can be contained in the $\delta$-neighborhood of a plane. A collection of tubes satisfies the polynomial Wolff axioms if not too many tubes can be contained in the $\delta$-neighborhood of a low degree algebraic variety. First, we prove that if a set of ${\delta }^{-3}$ tubes in ${\mathbb{R}}^4$ satisfies the polynomial Wolff axioms, then the union of the tubes must have volume at least ${\delta }^{1-1/40}$. We also prove a more technical statement which is analogous to a maximal function estimate at dimension $3+1/40$. Second, we prove that if a collection of ${\delta }^{-3}$ tubes in ${\mathbb{R}}^4$ 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 ${\delta }^{3/4}$. 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 $3+1/40$, and in particular this implies that every Kakeya set in ${\mathbb{R}}^4$ must have Hausdorff dimension at least $3+1/40$. This would be an improvement over the current best bound of 3, which was established by Wolff in 1995.

30 pages, 0 figures. v3: An error has been corrected, which weakens the dimension of our maximal function estimate to 3 + 1/40. Previously, a bound of 3 + 1/28 was claimed. See the addendum following the statement of Theorem 1.1 for details

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