Fourier dimension of constant rank hypersurfaces

TL;DR

This paper proves that the Fourier dimension of constant rank hypersurfaces in Euclidean space equals the number of non-vanishing principal curvatures, extending Harris's results on the light cone.

Contribution

It establishes the conjecture that Fourier dimension matches the count of non-vanishing principal curvatures for all constant rank hypersurfaces, generalizing previous specific cases.

Findings

Fourier dimension of constant rank hypersurfaces equals the number of non-vanishing principal curvatures.

Generalization of Harris's strategy to a broader class of hypersurfaces.

Confirmation of the conjecture for all 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.