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Fourier dimension of conical and cylindrical hypersurfaces
Classical Analysis and ODEs · Submitted 2 Jan 2024
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The notions of Hausdorff and Fourier dimensions are ubiquitous in harmonic analysis and geometric measure theory. | background 1.00 | 0.03 | 0.01 | asserted 1.00 |
| s1 | It is known that any hypersurface in $\mathbb{R}^{d+1}$ has Hausdorff dimension $d$. | background 1.00 | 0.03 | 0.02 | asserted 1.00 |
| s2 | However, the Fourier dimension depends on the finer geometric properties of the hypersurface. | background 0.98 | 0.04 | 0.02 | asserted 1.00 |
| s3 | For instance, the Fourier dimension of a hyperplane is 0, and the Fourier dimension of a hypersurface with non-vanishing Gaussian curvature is $d$. | background 1.00 | 0.04 | 0.02 | asserted 1.00 |
| s4 | 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. | background 0.87 | 0.03 | 0.05 | asserted 0.44 |
| s5 | 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. | result 0.57 | 0.97 | 0.93 | asserted 1.00 |
| s6 | In particular, cones and cylinders are not Salem. | result 0.73 | 0.88 | 0.15 | asserted 1.00 |
| s7 | Our method involves substantial generalizations of Harris's strategy. | method 0.98 | 0.86 | 0.03 | asserted 1.00 |
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Original 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.
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