论文标题
交叉分布，非命令指数和仿射平面中的Kakeya集
Intersection distribution, non-hitting index and Kakeya sets in affine planes
论文作者
论文摘要
我们提出了交叉分布和非命令指数的概念，可以从两个相关的角度来看。第一个涉及的是，在经典投影平面$ pg（2，q）$中的点$ s $ s $ s $ s $，其中$ s $的相交分布表示$ s $和$ pg（2，q）$的线之间的相交模式。第二个涉及有限字段$ \ mathbb {f} _q $上的多项式$ f $，其中$ f $的相交分布记录了polyenmials $ \ \ {f（x）+cx \ mid cx \ mid c \ in \ mathbb {f} _q \ _q \} $的总体分布属性。从某种意义上说，这两个观点是密切相关的，从某种意义上说，每个多项式都会以规范的方式产生$（q+1）$ - 设置，相反，每个$（q+1）$ - 设置了具有某些属性的每个属性具有多项式表示。实际上，基于相应$（q+1）$ sets的几何特性，相交分布提供了一个新的角度，可以在有限字段上区分多项式。在交叉分布中，我们确定了一个特别有趣的数量，称为非命中指数。对于点集$ s $，其非命令指数计算了$ pg（2，q）$中的线数，这些线路不会达到$ s $。对于有限字段$ \ mathbb {f} _q $上的多项式$ f $，其非命令索引给出了$ q $ value sets $ \ \ {f（x）+cx \ cx \ mid x \ in \ in \ mathbb {f} {f} _______________q \ c \ in $ c \ in $ c \ in $ c \ in的总和。我们在非命中索引上得出边界，并表明非命令索引包含有关相应集和多项式的大量信息。更确切地说，使用几何方法，我们表明，当非命中指数足以表征相应的点集和当多项式接近下限和上限时。此外，我们采用代数方法来得出几个点集和多项式家族的交点分布，并计算仿射平面中相关Kakeya集的大小。
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 $\{f(x)+cx \mid c \in \mathbb{F}_q\}$. 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 $\{f(x)+cx \mid x \in \mathbb{F}_q\}$, where $c \in \mathbb{F}_q$. We derive 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 close to the lower and upper bounds. Moreover, we employ an algebraic approach to derive the intersection distribution of several families of point sets and polynomials, and compute the sizes of related Kakeya sets in affine planes.
