论文标题
$ \ Mathbb f_q^2 $ in arcs
Arcs in $\mathbb F_q^2$
论文作者
论文摘要
ARC是$ \ Mathbb f_q^2 $的子集，不包含任何共线三倍。令$ a（q，k）$表示$ \ mathbb f_q^2 $中的弧数，带有基数$ k $。本文主要涉及估计$ a（q，k）$的大小时，$ k $相对较大，即$ k = q^t $对于某些$ t> 0 $。微不足道的估计告诉我们 \ [ {q \选择K} \ leq a（q，k）\ leq {q^2 \选择k}。 \] 我们表明，$ a（q，k）$的行为发生了显着接近$ t = 1/2 $。在此阈值之下，基本参数用于证明上述琐碎的上限无法显着改善。 另一方面，对于$ t \ geq 1/2+δ$，我们使用HyperGraph容器的理论获得改进的上限 \ [ a（q，k）\ leq {q^{2-t+2δ} \选择k}。 \] 该技术还用于在$ \ Mathbb f_q^2 $的随机子集中为最大弧的大小提供上限，该子集具有很高的概率。例如，我们证明A $ p $ -random子集$ q \ subset \ mathbb f_q^2 $带有$ q^{ - 3/2} <p <q^{ - 1} $包含一个大小$ω（q^{1/2}）的弧线，具有较高的可能性。结果对于$ p $的范围是最佳的。 最后，使用随机组中的ARC的最佳绑定用于证明Balogh和Solymosi结果的有限场类似物，具有更好的指数：存在一个子集$ p \ subset \ subset \ mathbb f_q^2 $，该\ Mathbb f_q^2 $不包含任何共线Quadruplace，但与每个$ P'\ comp p $ p $ p $ p $ p $ p' \ geq | p |^{3/4+o（1）} $，$ p'$包含一个共线三重。
An arc is a subset of $\mathbb F_q^2$ which does not contain any collinear triples. Let $A(q,k)$ denote the number of arcs in $\mathbb F_q^2$ with cardinality $k$. This paper is primarily concerned with estimating the size of $A(q,k)$ when $k$ is relatively large, namely $k=q^t$ for some $t>0$. Trivial estimates tell us that \[ {q \choose k} \leq A(q,k) \leq {q^2 \choose k}. \] We show that the behaviour of $A(q,k)$ changes significantly close to $t=1/2$. Below this threshold an elementary argument is used to prove that the trivial upper bound above cannot be improved significantly. On the other hand, for $t \geq 1/2+δ$, we use the theory of hypergraph containers to get an improved upper bound \[ A(q,k) \leq {q^{2-t+2δ} \choose k}. \] This technique is also used to give an upper bound for the size of the largest arc in a random subset of $\mathbb F_q^2$ which holds with high probability. For example, we prove that a $p$-random subset $Q \subset \mathbb F_q^2$ with $q^{-3/2}<p<q^{-1}$ contains an arc of size $Ω(q^{1/2})$ with high probability. The result is optimal for this range of $p$. Finally, this optimal bound for arcs in random sets is used to prove a finite field analogue of a result of Balogh and Solymosi, with a better exponent: there exists a subset $P \subset \mathbb F_q^2$ which does not contain any collinear quadruples, but with the property that for every $P' \subset P$ with $|P'| \geq |P|^{3/4+o(1)}$, $P'$ contains a collinear triple.
