Pi01 sets and tilings

In this paper, we prove that given any \Pi^0_1 subset $P$ of $\{0,1{\}}^{\text{NN}}$ there is a tileset $\tau$ with a set of configurations $C$ such that $P\times {\text{ZZ}}^2$ is recursively homeomorphic to $C\setminus U$ where $U$ is a computable set of configurations. As a consequence, if $P$ is countable, this tileset has the exact same set of Turing degrees.