Fourier analytic properties of Kakeya sets in finite fields
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912376086855680 |
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| author | Fraser, Jonathan M. |
| author_facet | Fraser, Jonathan M. |
| contents | We prove that a Kakeya set in a vector space over a finite field of size $q$ always supports a probability measure whose Fourier transform is bounded by $q^{-1}$ for all non-zero frequencies. We show that this bound is sharp in all dimensions at least 2. In particular, this provides a new and self-contained proof that a Kakeya set in dimension 2 has size at least $q^2/2$ (which is asymptotically sharp). We also establish analogous results for sets containing $k$-planes in a given set of orientations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_09464 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fourier analytic properties of Kakeya sets in finite fields Fraser, Jonathan M. Combinatorics Classical Analysis and ODEs Metric Geometry primary: 52C35, 43A25, secondary: 11B30, 52C17 We prove that a Kakeya set in a vector space over a finite field of size $q$ always supports a probability measure whose Fourier transform is bounded by $q^{-1}$ for all non-zero frequencies. We show that this bound is sharp in all dimensions at least 2. In particular, this provides a new and self-contained proof that a Kakeya set in dimension 2 has size at least $q^2/2$ (which is asymptotically sharp). We also establish analogous results for sets containing $k$-planes in a given set of orientations. |
| title | Fourier analytic properties of Kakeya sets in finite fields |
| topic | Combinatorics Classical Analysis and ODEs Metric Geometry primary: 52C35, 43A25, secondary: 11B30, 52C17 |
| url | https://arxiv.org/abs/2505.09464 |