A regular linear line complex is a three-parameter set of lines in space, whose Plücker vectors lie in a hyperplane, which is not tangent to the Klein quadric. Our main result is a bound $O(n^{1∕2}{m}^{3∕4}+m+n)$ for the number of incidences between $n$ lines in a complex and $m$ points in ${\mathbb{F}}^3$, where $\mathbb{F}$ is a field, and $n\le char{\left(\mathbb{F}\right)}^{4∕3}$ in positive characteristic. Zahl has recently observed that bichromatic pairwise incidences of lines coming from two distinct line complexes account for the nonzero single distance problem for a set of $n$ points in ${\mathbb{F}}^3$. This implied the new bound $O\left(n^{3∕2}\right)$ for the number of realisations of the distance, which is a square, for $\mathbb{F}$, where $-1$ is not a square in the $\mathbb{F}$-analogue of the Erdős single distance problem in ${\mathbb{R}}^3$. Our incidence bound yields, under a natural constraint, a weaker bound $O\left(n^{1.6}\right)$, which holds for any distance, including zero, over any $\mathbb{F}$.

规则的线性复线是空间中的三参数线集，其普吕克向量位于超平面中，该超平面与克莱因二次曲面不相切。我们的主要结果是一个界限$哦(n^{1∕2}{米}^{3∕4}+米+n)$ 之间的发生次数 $n$ 在复杂的线条和 $米$ 点在 ${\mathbb{F}}^3$， 在哪里 $\mathbb{F}$ 是一个字段，并且 $n\le CH\mathrm{一种}r{\left(\mathbb{F}\right)}^{4∕3}$在积极的特点。Zahl 最近观察到，来自两个不同线复合体的线的双色成对发生率解释了一组非零单距离问题$n$ 点在 ${\mathbb{F}}^3$. 这暗示了新的界限$哦\left(n^{3∕2}\right)$ 对于距离的实现次数，它是一个正方形，对于 $\mathbb{F}$， 在哪里 $-1$ 不是一个正方形 $\mathbb{F}$-Erdős 单距离问题的类比 ${\mathbb{电阻}}^3$. 我们的关联界在自然约束下产生较弱的界$哦\left(n^{1.6}\right)$，它适用于任何距离，包括零，在任何 $\mathbb{F}$.