Counting arcs in $\mathbb F_q^2$

TL;DR

This paper employs hypergraph container methods to derive upper bounds on the number of arcs in finite fields, providing near-tight estimates for various sizes and advancing combinatorial understanding of these geometric configurations.

Contribution

It introduces new upper bounds on the count of arcs in finite fields, improving previous results and applying hypergraph container techniques to geometric combinatorics.

Findings

Upper bound on total arcs: |A(q)| A0 A0 A0 A(q)| A0 A0 A0 A(q)| A0 A0 A0 A(q)

Upper bounds for arcs of fixed large size: |A(q,k)| A0 A0 A0 A(q,k)| A0 A0 A0 A(q,k)| A0 A0 A0 A(q,k)

Matching lower bounds are given by considering subsets of size k of an arc of size q.

Abstract

An arc in $\mathbb F_q^2$ is a set $P \subset \mathbb F_q^2$ such that no three points of $P$ are collinear. We use the method of hypergraph containers to prove several counting results for arcs. Let $\mathcal A(q)$ denote the family of all arcs in $\mathbb F_q^2$. Our main result is the bound \[ |\mathcal A(q)| \leq 2^{(1+o(1))q}. \] This matches, up to the factor hidden in the $o(1)$ notation, the trivial lower bound that comes from considering all subsets of an arc of size $q$. We also give upper bounds for the number of arcs of a fixed (large) size. Let $k=q^t$ for some $t >2/3$, and let $\mathcal A(q,k)$ denote the family of all arcs in $\mathbb F_q^2$ with cardinality $k$. We prove that, for all $\gamma >0$ \[ |\mathcal A(q,k)| \leq \binom{(1+\gamma)q}{k}. \] This result improves a bound of Roche-Newton and Warren. A nearly matching lower bound \[ |\mathcal A(q,k)| \geq \binom{q}{k} \] follows by considering all subsets of size $k$ of an arc of size $q$.