a suitable region $W$ of the Euclidean subspace $\mathbb{R}^n$ orthogonal to $\mathbb{R}^d$. The model set in question thereby emerges as the orthogonal projection in $\mathbb{R}^d$ of certain points of $\Sigma$, namely those with orthogonal projection in $\mathbb{R}^n$ falling into the region $W$. Mathematically a model set thus writes $$\begin{aligned}
 \Lambda _0:= \left\lbrace \gamma ^{\Vert } \; \vert \; \gamma \in \Sigma \; \text{ and } \; \gamma ^{\perp} \in W \right\rbrace\end{aligned}$$ where $^{\Vert }$ and $^{\perp}$ denotes the orthogonal projections onto $\mathbb{R}^d$ and $\mathbb{R}^n$ respectively. In the above context we will speak about a *real model set* (see the discussion of section $1$), the word “real” resulting from the fact that the summand $\mathbb{R}^n$ used here to form the cut $ \& $ project scheme is an Euclidean space.

The dynamical system $(\mathbb{X},\mathbb{R}^d)$ associated with a real model set is of very particular form: It is an *almost automorphic* extension over a torus $\mathbb{T}^{n+d}:=\mathbb{R}^{n+d}\diagup _{\Sigma }$ (see the material of section $2$). This property will prove to be central in our task, and shows up with the consideration of a certain factor map, also known as the *parametrization map* [@BaaHePl] [@Sch] $$\begin{aligned}
\begin{psmatrix}[colsep=1.5cm,
rowsep=1.2cm]
\mathbb{X} \; \;  & \; \; 