Flows on a cubic lattice (Q656375)

The following is shown about measure transportation on \({\mathbb Z}^n\): Let \(M \subseteq {\mathbb Z}^n\) be a set. Suppose that there is some constant \(C \in {\mathbb R}\) such that, for every positive integer \(l\), every hypercube in \({\mathbb R}^n\) with sides of length \(l\) and parallel to the coordinate axes contains at most \(Cl^{n-1}\) points from \({\mathbb Z}^n \setminus M\). Then there exists a surjective map \(f:M \rightarrow {\mathbb Z}^n\) and a constant \(K\) such that \(\|f(m)-m\| \leq K\) for all \(m \in M\).