Pi01 sets and tilings

In this paper, we prove that given any Π^0_1 subset $P$ of $\{0,1\}^\NN$ there is a tileset $τ$ with a set of configurations $C$ such that $P\times\ZZ^2$ 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.

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