论文标题
球体的角度，线路发生和Lie的线路通信
Sphere tangencies, line incidences, and Lie's line-sphere correspondence
论文作者
论文摘要
如果$ | p-q |^2 =（r-s）^2 $，则两个带有中心的球体$ p $和$ q $和签名的Radii $ r $和$ s $。使用Lie的线条通信，我们表明，如果$ f $是一个领域，其中$ -1 $不是正方形，那么$ f^3 $的一组球与适当构建的Heisenberg Group中的一组线之间存在同构，该线在$（f [i]）中嵌入了$（f [i]）^3 $;在这种同构中，球之间的接触转化为线之间的发病率。 在过去的十年中，了解三个空间中线的发生率几何形状取得了重大进展。接触率的同构允许我们将有关线的发生率几何形状的陈述转化为有关球体接触几何形状的陈述。这导致了ERDS在$ f^3 $中的重复距离问题的新界限，并在三个维度中提高了点状发生率的数量。对于某些参数范围，这些新界限很清晰。
Two spheres with centers $p$ and $q$ and signed radii $r$ and $s$ are said to be in contact if $|p-q|^2 = (r-s)^2$. Using Lie's line-sphere correspondence, we show that if $F$ is a field in which $-1$ is not a square, then there is an isomorphism between the set of spheres in $F^3$ and the set of lines in a suitably constructed Heisenberg group that is embedded in $(F[i])^3$; under this isomorphism, contact between spheres translates to incidences between lines. In the past decade there has been significant progress in understanding the incidence geometry of lines in three space. The contact-incidence isomorphism allows us to translate statements about the incidence geometry of lines into statements about the contact geometry of spheres. This leads to new bounds for Erdős' repeated distances problem in $F^3$, and improved bounds for the number of point-sphere incidences in three dimensions. These new bounds are sharp for certain ranges of parameters.
