Weak colored local rules for planar tilings

TL;DR

This paper characterizes linear subspaces with weak colored local rules as exactly those that are computable, extending previous algebraic-based results to a broader class of subspaces.

Contribution

It establishes a new characterization linking weak colored local rules to computability for linear subspaces, beyond algebraic cases.

Findings

Weak colored local rules exist if and only if the subspace is computable.

The characterization applies to sets of linear subspaces, including all of b^n.

Extends previous algebraic results to a more general computability framework.

Abstract

A linear subspace $E$ of $\mathbb{R}^n$ has colored local rules if there exists a finite set of decorated tiles whose tilings are digitizations of $E$. The local rules are 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 $\mathbb{R}^n$.