Counting Arcs in $${\mathbb {F}}_q^2$$

Pub Date : 2024-01-08 DOI:10.1007/s00454-023-00622-w

Krishnendu Bhowmick, Oliver Roche-Newton

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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

$$\begin{aligned} |{\mathcal {A}}(q)| \le 2^{(1+o(1))q}. \end{aligned}$$

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 $k \ge q^{2/3}(\log q)^3$, and let ${\mathcal {A}}(q,k)$ denote the family of all arcs in $\mathbb F_q^2$ with cardinality k. We prove that

$$\begin{aligned} |{\mathcal {A}}(q,k)| \le \left( {\begin{array}{c}(1+o(1))q\\ k\end{array}}\right) . \end{aligned}$$

This result improves a bound of Roche-Newton and Warren [12]. A nearly matching lower bound

$$\begin{aligned} |{\mathcal {A}}(q,k)| \ge \left( {\begin{array}{c}q\\ k\end{array}}\right) \end{aligned}$$

follows by considering all subsets of size k of an arc of size q.

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计算 $${mathbb {F}}_q^2$ 中的弧线

在 $\mathbb F_q^2$ 中的弧是一个集合 $P \subset \mathbb F_q^2$ ，使得 P 中没有三个点是相交的。我们使用超图容器的方法来证明弧的几个计数结果。让 ${\mathcal {A}}(q)$ 表示 $\mathbb F_q^2$ 中所有弧的族。我们的主要结果是约束 $$\begin{aligned}|2^{(1+o(1))q}.\end{aligned}$$这与考虑大小为 q 的弧的所有子集所得到的微不足道的下界相匹配，最多不超过 o(1) 符号中隐藏的因子。让 $k \ge q^{2/3}(\log q)^3$, 并让${\mathcal {A}}(q,k)$ 表示 $\mathbb F_q^2$ 中心智数为 k 的所有弧的族。|{\mathcal {A}}(q,k)| \le \left( {\begin{array}{c}(1+o(1))q\ k\end{array}\right) .\end{aligned}$$这个结果改进了罗切-牛顿和沃伦[12]的一个界限。一个几乎匹配的下界 $$\begin{aligned}|{mathcal {A}}(q,k)| |ge \left( {\begin{array}{c}q\ k\end{array}}\right) \end{aligned}$$通过考虑大小为 q 的弧的所有大小为 k 的子集，可以得出这个结果。

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