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

Rajko Nenadov

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.

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

Abstract

A subset of is called an arc if it does not contain three collinear points. We show that there are at most arcs of size , nearly matching a trivial lower bound . This was previously known to hold for , due to Bhowmick and Roche-Newton. The lower bound on is best possible up to a logarithmic factor.

Paper Structure

This paper contains 3 sections, 4 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1.1

For every $\varepsilon > 0$ there exists $C > 0$, such that the number of arcs in $\mathbb{F}_q^2$ of size $m \ge C q^{1/2} (\log q)^{3/2}$ is at most

Theorems & Definitions (6)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 3.1
  • proof
  • proof : Proof of Theorem \ref{['thm:main']}