In this paper, we propose the concepts of intersection distribution and non-hitting index, which can be viewed from two related perspectives. The first one concerns a point set S of size $q+1$ in the classical projective plane $PG(2,q)$, where the intersection distribution of S indicates the intersection pattern between S and the lines in $PG(2,q)$. The second one relates to a polynomial f over a finite field ${\mathbb{F}}_q$, where the intersection distribution of f records an overall distribution property of a collection of polynomials $\{f(x)+cx|c\in {\mathbb{F}}_q\}$. These two perspectives are closely related, in the sense that each polynomial produces a $(q+1)$-set in a canonical way and conversely, each $(q+1)$-set with certain property has a polynomial representation. Indeed, the intersection distribution provides a new angle to distinguish polynomials over finite fields, based on the geometric property of the corresponding $(q+1)$-sets. Among the intersection distribution, we identify a particularly interesting quantity named non-hitting index. For a point set S, its non-hitting index counts the number of lines in $PG(2,q)$ which do not hit S. For a polynomial f over a finite field ${\mathbb{F}}_q$, its non-hitting index gives the summation of the sizes of q value sets $\{f(x)+cx|x\in {\mathbb{F}}_q\}$, where $c\in {\mathbb{F}}_q$. We derive lower and upper bounds on the non-hitting index and show that the non-hitting index contains much information about the corresponding set and the polynomial. More precisely, using a geometric approach, we show that the non-hitting index is sufficient to characterize the corresponding point set and the polynomial, when it is very close to the lower and upper bounds. Moreover, we employ an algebraic approach to derive the non-hitting index and the intersection distribution of several families of point sets and polynomials. As an application, we consider the determination of the sizes of Kakeya sets in affine planes. The polynomial viewpoint of intersection distributions enable us to compute the size of a few families of Kakeya sets with nice algebraic properties. Finally, we describe the connection between these new concepts and various known results developed in different contexts and propose some future research problems.

在本文中，我们提出了交点分布和非击中指标的概念，可以从两个相关的角度来观察。第一个涉及大小为S的点集$q+\mathrm{1个}$ 在经典的投影平面上 $PG（2，q）$，其中S的相交分布表示S和其中的线之间的相交样式$PG（2，q）$。第二个涉及有限域上的多项式f${\mathbb{F}}_q$，其中f的交分布记录了多项式集合的整体分布特性$\{F（X）+CX|C\in {\mathbb{F}}_q\}$。这两个观点紧密相关，因为每个多项式都会产生一个$（q+\mathrm{1个}）$-以规范的方式设置，相反，每个 $（q+\mathrm{1个}）$具有某些属性的集合具有多项式表示形式。实际上，交点分布提供了一个新的角度，可以根据相应字段的几何特性来区分有限域上的多项式$（q+\mathrm{1个}）$集。在交点分布中，我们确定了一个特别有趣的数量，称为非击中索引。对于点集S，其非命中索引会计算点中的线数$PG（2，q）$不打小号。对于有限域上的多项式f${\mathbb{F}}_q$，其非命中索引可得出q个值集的大小之和$\{F（X）+CX|X\in {\mathbb{F}}_q\}$，在哪里 $C\in {\mathbb{F}}_q$。我们推导出非命中索引的上下限，并表明非命中索引包含有关相应集合和多项式的大量信息。更准确地说，使用几何方法，当非下标索引非常接近上下限时，它足以表征相应的点集和多项式。此外，我们采用代数方法得出点集和多项式的几个族的非命中指数和交点分布。作为一种应用，我们考虑确定仿射平面中的Kakeya集的大小。相交分布的多项式观点使我们能够计算具有良好代数性质的几套Kakeya集的大小。最后，