Local rules for multi-dimensional quasicrystals (Q1842130)

Let \(\mathbb{R}^{2k}\), \(k \geq 2\), be an Euclidean space equipped with an orthogonal base. Let \(E\) be a subspace of dimension \(k\) spanned by \(k\) vectors whose coordinates are in \(\mathbb{Q}[\sqrt{D}]\), where \(D\) is a positive integer. There is a set \({\mathcal T}_ E\) of quasi-periodic tilings associated with \(E\), obtained by the well-known strip projection procedure. We prove that this set \({\mathcal T}_ E\) always admit local rules which involve some coloring. This means that \({\mathcal T}_ E\) is exactly the set of all tilings satisfying the local rules.