Sets preserved by a large subgroup of the special linear group
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866913787082178560 |
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| author | Hung, Le Quang Pham, Thang Slavov, Kaloyan |
| author_facet | Hung, Le Quang Pham, Thang Slavov, Kaloyan |
| contents | Let $E$ be a subset of the affine plane over a finite field $\mathbb{F}_q$. We bound the size of the subgroup of $SL_2(\mathbb{F}_q)$ that preserves $E$. As a consequence, we show that if $E$ has size $\ll q^α$ and is preserved by $\gg q^β$ elements of $SL_2(\mathbb{F}_q)$ with $β\geq 3α/2$, then $E$ is contained in a line. This result is sharp in general, and will be proved by using combinatorial arguments and applying a point-line incidence bound in $\mathbb{F}_q^3$ due to Mockenhaupt and Tao (2004). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_01697 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sets preserved by a large subgroup of the special linear group Hung, Le Quang Pham, Thang Slavov, Kaloyan Combinatorics Classical Analysis and ODEs Group Theory Number Theory Let $E$ be a subset of the affine plane over a finite field $\mathbb{F}_q$. We bound the size of the subgroup of $SL_2(\mathbb{F}_q)$ that preserves $E$. As a consequence, we show that if $E$ has size $\ll q^α$ and is preserved by $\gg q^β$ elements of $SL_2(\mathbb{F}_q)$ with $β\geq 3α/2$, then $E$ is contained in a line. This result is sharp in general, and will be proved by using combinatorial arguments and applying a point-line incidence bound in $\mathbb{F}_q^3$ due to Mockenhaupt and Tao (2004). |
| title | Sets preserved by a large subgroup of the special linear group |
| topic | Combinatorics Classical Analysis and ODEs Group Theory Number Theory |
| url | https://arxiv.org/abs/2501.01697 |