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，若其成员均非自正交，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 $\delta = \delta(p)$，使得对任意特征为 $p$ 的域 $\mathbb{F}$ 以及任意整数 $k \geq 2$ 和 $d \geq k^{1/(p-1)}$，都存在至少 $d^{\delta \cdot k^{1/(p-1)}/ \log k}$ 个向量的 $\mathbb{F}^d$ 中 $k$-近正交集。特别地，在二元域上我们得到 $d^{\Omega( k /\log k)}$ 个向量的集合，且此结果在指数上除 $\log k$ 项外是紧的。相比之下，实数域上已知最佳下界为 $d^{\Omega( \log k / \log \log k)}$（Alon 与 Szegedy，Graphs and Combin.，1999）。该证明结合了概率方法与谱分析方法。