# Canonical projection tilings defined by patterns

## TL;DR

This paper establishes a precise algebraic condition under which certain affine subspaces can be uniquely characterized by finite pattern sets in their digitizations, linking algebraic properties with combinatorial tiling rules.

## Contribution

It provides a necessary and sufficient condition for affine subspaces to be characterized by finite forbidden pattern sets, connecting algebraic and combinatorial aspects of tilings.

## Key findings

- Only algebraic subspaces can be characterized by patterns.
- The condition relies on the notion of coincidence and is effectively checkable.
- Links algebraic properties of subspaces with combinatorics of digitizations.

## Abstract

We give a necessary and sufficient condition on a $d$-dimensional affine subspace of $\mathbb{R}^n$ to be characterized by a finite set of patterns which are forbidden to appear in its digitization. This can also be stated in terms of local rules for canonical projection tilings, or subshift of finite type. This provides a link between algebraic properties of affine subspaces and combinatorics of their digitizations. The condition relies on the notion of {\em coincidence} and can be effectively checked. As a corollary, we get that only algebraic subspaces can be characterized by patterns.