For a field $\mathbb{F}$ and integers $d$ and $k$, a set of vectors of $\mathbb{F}^d$ is called $k$-nearly orthogonal if its members are non-self-orthogonal and every $k+1$ of them include an orthogonal pair. We prove that for every prime $p$ there exists a positive constant $\delta = \delta (p)$, such that for every field $\mathbb{F}$ of characteristic $p$ and for all integers $k \geq 2$ and $d \geq k^{1/(p-1)}$, there exists a $k$-nearly orthogonal set of at least $d^{\delta \cdot k^{1/(p-1)}/ \log k}$ vectors of $\mathbb{F}^d$. In particular, for the binary field we obtain a set of $d^{\Omega( k /\log k)}$ vectors, and this is tight up to the $\log k$ term in the exponent. For comparison, the best known lower bound over the reals is $d^{\Omega( \log k / \log \log k)}$ (Alon and Szegedy, Graphs and Combin., 1999). The proof combines probabilistic and spectral arguments.

翻译：设$\mathbb{F}$为域，$d$和$k$为整数。$\mathbb{F}^d$中的一组向量称为$k$-近正交集，若其成员不自正交，且任意$k+1$个向量中必包含一对正交向量。我们证明：对每个素数$p$，存在正常数$\delta = \delta(p)$，使得对特征为$p$的任意域$\mathbb{F}$及所有满足$k \geq 2$和$d \geq k^{1/(p-1)}$的整数$d,k$，存在至少包含$d^{\delta \cdot k^{1/(p-1)}/ \log k}$个向量的$\mathbb{F}^d$中的$k$-近正交集。特别地，对二元域我们得到大小为$d^{\Omega( k /\log k)}$的向量集，该结果在指数项上紧致到$\log k$因子。相比之下，实数域上已知的最佳下界为$d^{\Omega( \log k / \log \log k)}$（Alon and Szegedy, Graphs and Combin., 1999定理）。该证明结合了概率与谱方法。