Steinhaus Sets and Jackson Sets
Logic
2007-05-23 v1 Metric Geometry
Abstract
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?
Cite
@article{arxiv.math/0603235,
title = {Steinhaus Sets and Jackson Sets},
author = {Su Gao and Arnold W. Miller and William A. R. Weiss},
journal= {arXiv preprint arXiv:math/0603235},
year = {2007}
}
Comments
Latex2e: 23 pages Latest version at http://www.math.wisc.edu/~miller