关于Erdos的直线交圆数目问题

On the Erdos' Problem of Numbers of Circles Intersected by Straights Lines

设U是平面E^2上任意给定的一组圆,L是E^2上与U中至少一个圆相交的直线的集合,X={x:存在l∈L使得x是原点O在l上的投影}.本文给出了极坐标测度不可忽略的点集的定义,并且证明了,当X是个这样的点集时,对任m>0及任p∈E^2,E^2中总存在着过p的直线与U中至少m个圆相交.本文还考虑了高维欧氏空间中λ维平面交n维球的数目等进一步的问题,并得到了一些结果.

Let U bc a family of any given circles, and L be a set of the straight lines in E^2 intersecting at least a circle in U. Let X={x: there is a linc l∈ L shch that x is the pro- jcction of the origin O onto l}. A difinition of point sets with non-negligible polar coordinates measures is given, and it is proved that if X is such a set then for any given m>0 and any p∈ E^2 there exists a straight line through p intersecting at least m circles in U. Moreover, some further problems are considered, for example, on numbers of n-dimensional balls intersected by λ-dimensional planes in higher-dimersional Euclidean spaces, etc. Some results of these problems are obtained.