A discretized Severi-type theorem with applications to harmonic analysis

Abstract

In 1901, Severi proved that if $Z$ is an irreducible hypersurface in ${\mathbb{P}}^4(\mathbb{C})$ 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 ${\delta }^{-2-\epsilon }$ 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 ${\mathbb{R}}^4$.

58 pages, 4 figures. v3: Typo corrected in Section 2 to match published version. Added journal-ref and DOI

References in corpus (2)

• Polynomial Wolff axioms and Kakeya-type estimates in ${\mathbb{R}}^4$
• An improved bound on the Hausdorff dimension of Besicovitch sets in ${\mathbb{R}}^3$

Cited by in corpus (3)

• New Kakeya estimates using Gromov's algebraic lemma
• A note on Fourier restriction and nested Polynomial Wolff axioms
• Fourier restriction in low fractal dimensions