Geometric and Functional Analysis

, Volume 28, Issue 4, pp 1131–1181 | Cite as

A discretized Severi-type theorem with applications to harmonic analysis

  • Joshua ZahlEmail author


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 \({\mathbb{R}^{4}}\).


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Authors and Affiliations

  1. 1.University of British ColumbiaVancouverCanada

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