A discretized Severi-type theorem with applications to harmonic analysis

TL;DR

This paper proves a discretized Severi-type theorem in , leading to improved bounds on Kakeya and restriction problems, and establishing a new lower bound on the Hausdorff dimension of Besicovitch sets.

Contribution

It introduces a discretized Severi-type theorem in  and applies it to improve bounds in Kakeya and restriction problems, and to Hausdorff dimension estimates.

Findings

Bound on  tubes in  neighborhoods of low-degree hypersurfaces.

Improved Kakeya maximal function estimate at dimension 3+1/28.

Hausdorff dimension of Besicovitch sets in  is at least 3+1/28.

Abstract

In 1901, Severi proved that if $Z$ is an irreducible hypersurface in $\mathbb{P}^4(\mathbb{C})$ that contains a three dimensional set of lines, then $Z$ is either a quadratic hypersurface or a scroll of planes. We prove a discretized version of this result for hypersurfaces in $\mathbb{R}^4$. As an application, we prove that at most $\delta^{-2-\varepsilon}$ direction-separated $\delta$-tubes can be contained in the $\delta$-neighborhood of a low-degree hypersurface in $\mathbb{R}^4$. This result leads to improved bounds on the restriction and Kakeya problems in $\mathbb{R}^4$. Combined with previous work of Guth and the author, this result implies a Kakeya maximal function estimate at dimension $3+1/28$, which is an improvement over the previous bound of $3$ due to Wolff. As a consequence, we prove that every Besicovitch set in $\mathbb{R}^4$ must have Hausdorff dimension at least $3+1/28$. Recently, Demeter showed that any improvement over Wolff's bound for the Kakeya maximal function yields new bounds on the restriction problem for the paraboloid in $\mathbb{R}^4$.