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

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 $\left(\genfrac{}{}{0ex}{}{(1+o(1))q}{m}\right)$ arcs of size $m\gg q^{1/2}(\log q{)}^{3/2}$, nearly matching a trivial lower bound $\left(\genfrac{}{}{0ex}{}{q}{m}\right)$. 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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