Given a set of points $P$ and a set of regions $\mathcal{O}$, an incidence is a pair $(p,o ) \in P \times \mathcal{O}$ such that $p \in o$. We obtain a number of new results on a classical question in combinatorial geometry: What is the number of incidences (under certain restrictive conditions)? We prove a bound of $O\bigl( k n(\log n/\log\log n)^{d-1} \bigr)$ on the number of incidences between $n$ points and $n$ axis-parallel boxes in $\mathbb{R}^d$, if no $k$ boxes contain $k$ common points, that is, if the incidence graph between the points and the boxes does not contain $K_{k,k}$ as a subgraph. This new bound improves over previous work, by Basit, Chernikov, Starchenko, Tao, and Tran (2021), by more than a factor of $\log^d n$ for $d >2$. Furthermore, it matches a lower bound implied by the work of Chazelle (1990), for $k=2$, thus settling the question for points and boxes. We also study several other variants of the problem. For halfspaces, using shallow cuttings, we get a linear bound in two and three dimensions. We also present linear (or near linear) bounds for shapes with low union complexity, such as pseudodisks and fat triangles.

翻译：给定点集$P$和区域集$\mathcal{O}$，若点$p \in P$落在区域$o \in \mathcal{O}$内，则称$(p, o)$为一个关联对。本文针对组合几何中的经典问题（特定约束条件下的关联数）取得若干新结果。我们证明了$\mathbb{R}^d$中$n$个点与$n$个轴平行盒之间的关联数上界为$O\bigl( k n(\log n/\log\log n)^{d-1} \bigr)$，其约束条件为：不存在$k$个盒同时包含$k$个公共点，即点与盒之间的关联图不含$K_{k,k}$子图。该新界比Basit、Chernikov、Starchenko、Tao和Tran（2021）的先前工作在$d>2$时改善了超过$\log^d n$倍。对于$k=2$的情况，该界与Chazelle（1990）隐含的下界匹配，从而解决了点与盒的关联数问题。此外，我们还研究了该问题的若干变体。对于半空间，利用浅层切割方法，我们在二维和三维空间中得到了线性界。对于具有低并复杂性的几何形状（如伪圆盘和胖三角形），我们也给出了线性（或近线性）界。