Steinhaus集和Jackson集
Steinhaus Sets and Jackson Sets
摘要：
我们证明，不存在一个子集的平面S，满足每一个等距副本的顶点的单位正方形在一个点。我们给出了实数的所有三个点子集F的一个完整的特征，使得不存在一个实数集S满足F的每个等距副本恰好在一个点上。平面中的有限集X是Jackson当且仅当对于平面的每个子集S，存在X的等距副本Y，使得Y不恰好在一点上与S相交。这些结果与公开问题Q有关。（史蒂夫·杰克逊）平面上的每个有限集X都是由两个或多个点组成的吗？
英文摘要：
We prove that there does not exist a subset of the plane S that meets every isometric copy of the vertices of the unit square in exactly one point. We give a complete characterization of all three point subsets F of the reals such that there does not exists a set of reals S which meets every isometric copy of F in exactly one point. A finite set X in the plane is Jackson iff for every subset S of the plane there exists an isometric copy Y of X such that Y does not meets S in exactly one point. These results are related to the open problem: Q. (Steve Jackson) Is every finite set X in the plane of two or more points Jackson?
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
