Fourier dimension of constant rank hypersurfaces

Abstract

Any hypersurface in ${\mathbb{R}}^{d+1}$ has a Hausdorff dimension of $d$. However, the Fourier dimension depends on the finer geometric properties of the hypersurface. For example, the Fourier dimension of a hyperplane is 0, and the Fourier dimension of a hypersurface with non-vanishing Gaussian curvature is $d$. Recently, Harris showed that the Euclidean light cone in ${\mathbb{R}}^{d+1}$ has a Fourier dimension of $d-1$, which leads one to conjecture that the Fourier dimension of a hypersurface equals the number of non-vanishing principal curvatures. We prove this conjecture for all constant rank hypersurfaces. Our method involves substantial generalizations of Harris's strategy.

22 pages, 3 figures