球面切触、直线关联与Lie的直线-球面对应
Sphere tangencies, line incidences, and Lie's line-sphere correspondence
AI总结：
该研究利用Lie的直线-球面对应建立域上球面与Heisenberg群内直线的同构，将直线关联几何结论转化为球面接触几何结论，得到Erdős重复距离问题新界与三维点-球面关联改进界。
摘要：
若球心为$p$、$q$且带符号半径为$r$、$s$的两个球面满足$|p-q|^2 = (r-s)^2$，则称二者处于接触状态。利用Lie的直线-球面对应，我们证明：若$F$是一个$-1$不为平方元的域，则$F^3$中的球面集合与嵌入在$(F[i])^3$中的一个适当构造的Heisenberg群（海森堡群）内的直线集合之间存在同构；在该同构下，球面间的接触对应于直线间的关联。\n过去十年间，三维空间中直线的关联几何研究取得了显著进展。这种接触-关联同构使我们能将关于直线关联几何的结论转化为关于球面接触几何的结论。这为$F^3$中的Erdős重复距离问题带来了新的界，并改进了三维中点-球面关联数量的界。这些新界在特定参数范围内是紧的。
英文摘要：
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.
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