Weak colored local rules for planar tilings

Thomas Fernique, Mathieu Sablik

Published 2016-03-31Version 1

A linear subspace $E$ of $\mathbb{R}^n$ has {\em colored local rules} if there exists a finite set of decorated tiles whose tilings are digitizations of $E$. The local rules are {\em weak} if the digitizations can slightly wander around $E$. We prove that a linear subspace has weak colored local rules if and only if it is computable. This goes beyond the previous results, all based on algebraic subspaces. We prove an analogous characterization for sets of linear subspaces, including the set of all the linear subspaces of $\mathbb{R}^n$.