Fourier dimension of constant rank hypersurfaces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929629639475200 |
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| author | Zhu, Junjie |
| author_facet | Zhu, Junjie |
| contents | 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, Harris showed 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 constant rank hypersurfaces. Our method involves substantial generalizations of Harris's strategy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_09711 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fourier dimension of constant rank hypersurfaces Zhu, Junjie Classical Analysis and ODEs 42B10, 42B20, 28A12, 53A07 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, Harris showed 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 constant rank hypersurfaces. Our method involves substantial generalizations of Harris's strategy. |
| title | Fourier dimension of constant rank hypersurfaces |
| topic | Classical Analysis and ODEs 42B10, 42B20, 28A12, 53A07 |
| url | https://arxiv.org/abs/2410.09711 |