Improved bounds for the Kakeya maximal conjecture in higher dimensions

Abstract

We adapt Guth's polynomial partitioning argument for the Fourier restriction problem to the context of the Kakeya problem. By writing out the induction argument as a recursive algorithm, additional multiscale geometric information is made available. To take advantage of this, we prove that direction-separated tubes satisfy a multiscale version of the polynomial Wolff axioms. Altogether, this yields improved bounds for the Kakeya maximal conjecture in ${\mathbb{R}}^n$ with $n=5$ or $n\ge 7$ and improved bounds for the Kakeya set conjecture for an infinite sequence of dimensions.

arXiv comment43 pages, 6 figures. This article supersedes arXiv:1901.01802

References in corpus (4)

• On the size of Kakeya sets in finite fields
• An improved bound on the Hausdorff dimension of Besicovitch sets in ${\mathbb{R}}^3$
• A restriction estimate in ${\mathbb{R}}^3$ using brooms
• A Kakeya maximal function estimate in four dimensions using planebrushes

Cited by in corpus (6)

• A note on Fourier restriction and nested Polynomial Wolff axioms
• On the Fourier dimension of $(d,k)$-sets and Kakeya sets with restricted directions
• Tomography bounds for the Fourier extension operator and applications
• New bounds for Stein's square functions in higher dimensions
• Maximal subspace averages
• Planar incidences and geometric inequalities in the Heisenberg group