Tilings of amenable groups

We prove that for any infinite countable amenable group $G$, any $\epsilon > 0$andanyfinitesubset$K\subset G$,thereexistsatiling(partitionof$G$intofinite\mathrm{"}tiles\mathrm{"}usingonlyfinitelymany\mathrm{"}shapes\mathrm{"}),whereallthetilesare$(K; \epsilon)$-invariant. Moreover, our tiling has topological entropy zero (i.e., subexponential complexity of patterns). As an application, we construct a free action of $G$ (in the sense that the mappings, associated to different from unity elements of $G$, have no fixpoints), on a zero-dimensional space, and which has topological entropy zero.