BRK-type sets over finite fields
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866910389822816256 |
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| author | Trainor, Charlotte |
| author_facet | Trainor, Charlotte |
| contents | A Besicovitch-Rado-Kinney (BRK) set in $\mathbb{R}^n$ is a Borel set that contains a $(n-1)$-dimensional sphere of radius $r$, for each $r>0$. It is known that such sets have Hausdorff dimension $n$ from the work of Kolasa and Wolff. In this paper, we consider an analogous problem over a finite field, $\mathbb{F}_q$. We define BRK-type sets in $\mathbb{F}_q^n$, and establish lower bounds on the size of such sets using techniques introduced by Dvir's proof of the finite field Kakeya conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_19824 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | BRK-type sets over finite fields Trainor, Charlotte Combinatorics 05B25, 11T99 A Besicovitch-Rado-Kinney (BRK) set in $\mathbb{R}^n$ is a Borel set that contains a $(n-1)$-dimensional sphere of radius $r$, for each $r>0$. It is known that such sets have Hausdorff dimension $n$ from the work of Kolasa and Wolff. In this paper, we consider an analogous problem over a finite field, $\mathbb{F}_q$. We define BRK-type sets in $\mathbb{F}_q^n$, and establish lower bounds on the size of such sets using techniques introduced by Dvir's proof of the finite field Kakeya conjecture. |
| title | BRK-type sets over finite fields |
| topic | Combinatorics 05B25, 11T99 |
| url | https://arxiv.org/abs/2403.19824 |