Abstract:

The Steinhaus lattice problem in ${\mathbb{R}}^n$ is whether there is a set $S \subseteq {\mathbb{R}}^n$ with the property that for every isometric copy $L$ of the integer lattice ${\mathbb{Z}}^n$ we have $|L \cap S|=1$. In 2000, Dr. Jackson Dr. Mauldin showed that for n=2 such sets do exist, and it is known that for $n\geq 4$ that they do not exist (for the standard lattice). The question for $n=3$ remains open. We present several results related to the $n=3$ version of the problem and several problems of interest in their own right which have arised in conjuction with this problem.