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arXiv · 2410.21818

The number of arcs in $\mathbb{F}_q^2$ of a given cardinality

Rajko Nenadov
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Abstract

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.

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Rajko Nenadov. 2024-10-29. The number of arcs in $\mathbb{F}_q^2$ of a given cardinality. https://arxiv.org/abs/2410.21818

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