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

TL;DR

This paper establishes an upper bound on the number of large arcs in finite fields, nearly matching the trivial lower bound, and improves previous bounds for the size of such arcs.

Contribution

It provides a nearly tight upper bound on the number of arcs of a given size in finite fields, extending previous results to smaller arc sizes.

Findings

Upper bound on the number of arcs of size m in _q^2

Improved bounds for arcs with size  m  q^{1/2} (\u221elog q)^{3/2}

Bounds are nearly optimal up to logarithmic factors

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.