Fourier dimension of conical and cylindrical hypersurfaces

TL;DR

This paper proves that the Fourier dimension of certain conical and cylindrical hypersurfaces in Euclidean space equals the number of their non-vanishing principal curvatures, extending Harris's results to a broader class of shapes.

Contribution

It establishes the Fourier dimension for all $d$-dimensional cones and cylinders generated by hypersurfaces with non-vanishing Gaussian curvature, confirming a conjecture for these cases.

Findings

Fourier dimension of cones and cylinders matches the count of non-vanishing principal curvatures.

Cones and cylinders are shown not to be Salem sets.

Method extends Harris's strategy to a wider class of hypersurfaces.

Abstract

The notions of Hausdorff and Fourier dimensions are ubiquitous in harmonic analysis and geometric measure theory. It is known that any hypersurface in $\mathbb{R}^{d+1}$ has Hausdorff dimension $d$. However, the Fourier dimension depends on the finer geometric properties of the hypersurface. For instance, the Fourier dimension of a hyperplane is 0, and the Fourier dimension of a hypersurface with non-vanishing Gaussian curvature is $d$. Recently, Harris has shown that the Euclidean light cone in $\mathbb{R}^{d+1}$ has Fourier dimension $d-1$, 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 $d$-dimensional cones and cylinders in $\mathbb{R}^{d+1}$ generated by hypersurfaces in $\mathbb{R}^d$ with non-vanishing Gaussian curvature. In particular, cones and cylinders are not Salem. Our method involves substantial generalizations of Harris's strategy.