A discretized Severi-type theorem with applications to harmonic analysis

Abstract

In 1901, Severi proved that if ${{Z}}$ is an irreducible hypersurface in ${\mathbb{P}^{4}(C)}$ 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 ${\delta^{{-2}{-\epsilon}}}$ direction-separated δ-tubes can be contained in the δ-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 ${\mathbb{R}^{4}}$.