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 F3 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 F3, and improved bounds for the number of point-sphere incidences in three dimensions. These new bounds are sharp for certain ranges of parameters.

中文翻译：

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球切线、线入射和李的线球对应

有中心的两个球体p和q和签名半径r和s据说有联系如果|p–q|2=(r–s)2. 使用李的线球对应，我们证明如果F是一个域，其中 –1 不是正方形，则在其中的球体集合之间存在同构F3以及嵌入在 (F[一世])3; 在这种同构下，球体之间的接触转化为线之间的入射。在过去的十年中，在理解三个空间中线的入射几何方面取得了重大进展。接触入射同构允许我们将关于线的入射几何的陈述翻译成关于球体的接触几何的陈述。这导致 Erdős 的重复距离问题在F3，并改进了三个维度上的点球入射数的界限。对于某些参数范围，这些新界限非常明显。