The number of arcs in $\mathbb{F}_q^2$ of a given cardinality
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Accesso online: | |
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| _version_ | 1866914996835844096 |
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| author | Nenadov, Rajko |
| author_facet | Nenadov, Rajko |
| contents | A subset of $\mathbb{F}_q^2$ is called an arc if it does not contain three collinear points. We show that there are at most $\binom{(1 + o(1))q}{m}$ arcs of size $m \gg q^{1/2} (\log q)^{3/2}$, nearly matching a trivial lower bound $\binom{q}{m}$. This was previously known to hold for $m \gg q^{2/3} (\log q)^3$, due to Bhowmick and Roche-Newton. The lower bound on $m$ is best possible up to a logarithmic factor. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_21818 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The number of arcs in $\mathbb{F}_q^2$ of a given cardinality Nenadov, Rajko Combinatorics A subset of $\mathbb{F}_q^2$ is called an arc if it does not contain three collinear points. We show that there are at most $\binom{(1 + o(1))q}{m}$ arcs of size $m \gg q^{1/2} (\log q)^{3/2}$, nearly matching a trivial lower bound $\binom{q}{m}$. This was previously known to hold for $m \gg q^{2/3} (\log q)^3$, due to Bhowmick and Roche-Newton. The lower bound on $m$ is best possible up to a logarithmic factor. |
| title | The number of arcs in $\mathbb{F}_q^2$ of a given cardinality |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2410.21818 |