A Simple Condition for Bounded Displacement

TL;DR

This paper provides a straightforward criterion based on eigenvalues of the substitution matrix to determine if separated nets from substitution tilings are a bounded displacement of the integer lattice.

Contribution

It introduces a simple condition involving eigenvalues and eigenspaces to identify when separated nets are bounded displacements of the integer lattice.

Findings

The condition effectively distinguishes bounded displacement cases.

Eigenvalues of the substitution matrix are key to the criterion.

The approach simplifies analysis of separated nets in Euclidean space.

Abstract

We study separated nets that correspond to substitution tilings of the Euclidean space. We give a simple condition, in terms of the eigenvalues and eigenspaces of the substitution matrix, to know whether the separated net is a bounded displacement of the integer lattice or not.