Pi01 sets and tilings

TL;DR

This paper demonstrates how to construct tilesets that encode any given _1 subset of infinite binary sequences into configurations, preserving Turing degrees for countable sets, linking tiling theory with computability.

Contribution

It introduces a method to represent any _1 subset of ^ as tiling configurations, establishing a connection between tilings and computability theory.

Findings

Constructed tilesets for any _1 subset of ^.

Configurations are recursively homeomorphic to the product of the subset and ^2.

For countable sets, the tileset preserves the exact set of Turing degrees.

Abstract

In this paper, we prove that given any \Pi^0_1 subset $P$ of $\{0,1\}^\NN$ there is a tileset $\tau$ 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.