{
  "id": "2401.01455",
  "version": "v1",
  "published": "2024-01-02T22:39:35.000Z",
  "updated": "2024-01-02T22:39:35.000Z",
  "title": "Fourier dimension of conical and cylindrical hypersurfaces",
  "authors": [
    "Junjie Zhu"
  ],
  "categories": [
    "math.CA"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-02T22:39:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B10",
      "42B20",
      "28A12",
      "53A05"
    ],
    "keywords": [
      "fourier dimension",
      "cylindrical hypersurfaces",
      "non-vanishing gaussian curvature",
      "euclidean light cone",
      "geometric measure theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}