Fourier dimension of conical and cylindrical 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.

Figures (3)

• Figure 1: Illustration of the re-parametrization by $T_t$ on $C_{1}$
• Figure 2: Illustration of the shift $\Theta_t$ on $C_{2}$, where $x_t=T_t(x)$.
• Figure 3: Illustration of the re-parametrization by $T_t$ on $D_{1}$

Theorems & Definitions (46)

• Conjecture 1.1
• Theorem 1.2
• Remark
• Theorem 1.3
• Proposition 2.1
• Proposition 2.2
• Proposition 2.3
• Proposition 2.4
• Remark
• proof : Proof of Proposition \ref{['p31']}