The notions of Hausdorff and Fourier dimensions are ubiquitous in harmonic analysis and geometric measure theory. Many questions on how the notions of dimensions relate to the geometry of a set remain unanswered. It is known that 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, Fraser, Harris, and Kroon have shown 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 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 the strategy by Fraser, Harris, and Kroon.
