For a field $\mathbb{F}$ and integers $d$ and $k$, a set ${\cal A} \subseteq \mathbb{F}^d$ is called $k$-nearly orthogonal if its members are non-self-orthogonal and every $k+1$ vectors of ${\cal A}$ include an orthogonal pair. We prove that for every prime $p$ there exists some $\delta = \delta(p)>0$, such that for every field $\mathbb{F}$ of characteristic $p$ and for all integers $k \geq 2$ and $d \geq k$, there exists a $k$-nearly orthogonal set of at least $d^{\delta \cdot k/\log k}$ vectors of $\mathbb{F}^d$. The size of the set is optimal up to the $\log k$ term in the exponent. We further prove two extensions of this result. In the first, we provide a large set ${\cal A}$ of non-self-orthogonal vectors of $\mathbb{F}^d$ such that for every two subsets of ${\cal A}$ of size $k+1$ each, some vector of one of the subsets is orthogonal to some vector of the other. In the second extension, every $k+1$ vectors of the produced set ${\cal A}$ include $\ell+1$ pairwise orthogonal vectors for an arbitrary fixed integer $1 \leq \ell \leq k$. The proofs involve probabilistic and spectral arguments and the hypergraph container method.

翻译：对于域$\mathbb{F}$及整数$d$和$k$，集合${\cal A} \subseteq \mathbb{F}^d$被称为$k$-近正交集，若其成员非自正交，且${\cal A}$中任意$k+1$个向量包含一对正交向量。我们证明：对每个素数$p$，存在某个$\delta = \delta(p)>0$，使得对任意特征为$p$的域$\mathbb{F}$及所有整数$k \geq 2$和$d \geq k$，存在一个至少包含$d^{\delta \cdot k/\log k}$个向量的$k$-近正交子集于$\mathbb{F}^d$中。该集合的大小在指数项上最优，仅相差$\log k$因子。我们进一步证明了该结果的两个推广。第一个推广中，我们构造了$\mathbb{F}^d$中一个由非自正交向量组成的大集合${\cal A}$，使得对于${\cal A}$的任意两个大小均为$k+1$的子集，其中一个子集的某个向量与另一个子集的某个向量正交。第二个推广中，对任意固定整数$1 \leq \ell \leq k$，所得集合${\cal A}$中任意$k+1$个向量均包含$\ell+1$个两两正交的向量。证明方法涉及概率与谱分析论证，以及超图容器方法。