2.1 Delone sets

A subset $\Lambda $ of $\mathbb {R}^d$ is called a Delone set if it is uniformly discrete, meaning that there is $r>0$ such that every closed ball of radius r intersects $\Lambda $ in at most one point; and relatively dense, which means that there is $R>0$ such that every closed ball of radius R intersects $\Lambda $ in at least one point.

Let $\Lambda $ be a Delone set in $\mathbb {R}^d$ . For every $t\in \mathbb {R}^d$ , we denote by $\Lambda -t$ the Delone set $\{x-t \mid x\in \Lambda \}$ .

For every $\rho>0$ and every t in $\mathbb {R}^d$ denote by $B(t,\rho )$ the open ball in $\mathbb {R}^d$ of radius $\rho $ and center t. A $\rho $ -patch of $\Lambda $ centered at $t\in \mathbb {R}^d$ is the set $\Lambda \cap \overline {B(t,\rho )}$ . We consider two notions of long-range order for Delone sets: the first states that a Delone set $\Lambda $ has finite local complexity if for every $\rho>0$ it has a finite number of $\rho $ -patches up to translation; and the second says that $\Lambda $ is repetitive if for each $\rho>0$ there is a number $M>0$ such that each closed ball of radius M contains the center of a translated copy of every possible $\rho $ -patch of $\Lambda $ . Observe that every repetitive Delone set has finite local complexity; see [Reference Baake and GrimmBG13, Proposition $5.6$ ].

2.2 Meyer sets and address map

Let $\Lambda $ be a Delone set in $\mathbb {R}^d$ . We say that $\Lambda $ is a Meyer set if there is a finite set F in $\mathbb {R}^d$ such that

$$ \begin{align*} \Lambda-\Lambda\subseteq \Lambda + F. \end{align*} $$

In [Reference MeyerMey72], Meyer proved that every model set is a Meyer set. The following characterization of Meyer set is used in the proofs of Proposition 1.1 and the main theorem.

Theorem 2.1. [Reference LagariasLag99, Theorem 3.1]

Let $\Lambda $ be a finitely generated Delone set in $\mathbb {R}^d$ with rank s. Then $\Lambda $ is a Meyer set if and only if every address map

$$ \begin{align*} \varphi: \langle \Lambda \rangle \to \mathbb{Z}^s, \end{align*} $$

is almost linear, that is, there are a unique linear map $\ell :\mathbb {R}^d\to \mathbb {R}^s$ and a constant $C>0$ such that for every x in $\Lambda $ we have

(2.1) $$ \begin{align} \| \varphi(x) - \ell(x) \|_s \le C. \end{align} $$

Remark 2.1. In the proof of [Reference LagariasLag99, Theorem 3.1] it was proved that $\ell $ is some kind of ‘ideal address map’ in the sense that if $\{v_1,\ldots , v_s\}$ is the basis of $\langle \Lambda \rangle $ that we used to define the address map of $\Lambda $ then for every t in $\mathbb {R}^d$ we have

(2.2) $$ \begin{align} \sum_{i=1}^s \ell_i(t) v_i = t. \end{align} $$

2.3 Dynamical systems and transverse groupoid

Let $\Lambda \subseteq \mathbb {R}^d$ be a Delone set with finite local complexity. The hull of $\Lambda $ is the collection of all Delone sets in $\mathbb {R}^d$ whose $\rho $ -patches, for every $\rho>0$ , are also $\rho $ -patches of $\Lambda $ up to translation. We denote this set by $\Omega _\Lambda $ . There is a natural metrizable topology on $\Omega _\Lambda $ . Roughly speaking, two Delone sets are close in this topology if they agree on a large ball around the origin up to a small translation. In particular, for every $\Lambda '$ in $\Omega _\Lambda $ a basis of open neighborhoods for $\Lambda '$ is given by the following sets. First, for every $R>0$ put

and for every $0<\varepsilon <R/2$ we define the open neighborhood $N(\Lambda ',\varepsilon , R)$ of $\Lambda '$ by

for more details see, for example, [Reference Forrest, Hunton and KellendonkFHK02, Reference Kellendonk and LenzKL13, Reference Lee and MoodyLM06, Reference SchlottmannSch00]. If $\Lambda $ has finite local complexity then its hull $\Omega _\Lambda $ is compact. Observe that the action by translation of $\mathbb {R}^d$ on $\Omega _\Lambda $ is continuous. Thus, we obtain a topological dynamical system denote by $(\Omega _\Lambda ,\mathbb {R}^d)$ . The orbit of x in $\Omega _\Lambda $ is the set $\{x-t \mid t\in \mathbb {R}^d\}$ , and a subset A of $\Omega _\Lambda $ is called invariant if it is invariant by the action of $\mathbb {R}^d$ . The dynamical system $(\Omega _\Lambda ,\mathbb {R}^d) $ is minimal if and only if the only closed invariant sets are the empty set and the whole space. It is well known that minimality is equivalent to the fact every point has a dense orbit, and in the context of Delone sets repetitivity is equivalent to minimality.

We recall that every topological dynamical system admits a maximal equicontinuous factor, that is, a topological factor with an equicontinuous action such that any other equicontinuous factor is a topological factor of it; see, for instance, [Reference Barge, Kellendonk and SchmiedingBKS12, Reference Barge and KellendonkBK13, Reference KurkaKur03]. For a topological dynamical system $(X,G)$ , where X is a compact metric space and G a locally compact abelian group, we denote by $(X_{\mathrm {me}},G)$ its maximal equicontinuous factor. Given two minimal dynamical systems $(X,G)$ and $(Y,G)$ , and a factor map $\pi :(X,G)\to (Y,G)$ , we say that $\pi :(X,G)\to (Y,G)$ is an almost automorphic extension or that $(X,G)$ is an almost automorphic extension of $(Y,G)$ if there is a point in Y with a unique preimage under $\pi $ .

The transversal of the hull is the closed subset

$$ \begin{align*}\Xi_{\Lambda}:=\{x\in\Omega_\Lambda\mid 0\in x\}\subseteq\Omega_\Lambda.\end{align*} $$

In general, the restriction of the action of $\mathbb {R}^d$ to $\Xi _\Lambda $ is not defined. For this reason, to study the dynamical properties of the transversal we introduce the transverse groupoid,

$$ \begin{align*}\mathfrak{G}_\Lambda=\{(x,t)\in\Xi_{\Lambda}\times\mathbb{R}^d\mid x-t\in\Xi_{\Lambda}\}\subseteq\Xi_{\Lambda}\times\mathbb{R}^d.\end{align*} $$

This set, endowed with the induced topology from the product space $\Xi _{\Lambda }\times \mathbb {R}^d$ , has the structure of a topological groupoid; see [Reference RenaultRen80] for the abstract definition of topological groupoids. Two elements $(x,t)$ and $(z,s)$ in $\mathfrak {G}_\Lambda $ are composable if and only if $x-t=z$ , and the composition of $(x,t)$ and $(z,s)$ is defined by

$$ \begin{align*}(x,t)\cdot(z,s)=(x,t+s).\end{align*} $$

The inverse map ${\cdot }^{-1}:\mathfrak {G}_\Lambda \to \mathfrak {G}_\Lambda $ is defined by $(x,t)^{-1}=(x-t,-t)$ and the domain $d:\mathfrak {G}_\Lambda \to \Xi _{\Lambda }$ and range $r:\mathfrak {G}_\Lambda \to \Xi _{\Lambda }$ maps are defined by

$$ \begin{align*}d(x,t)=x\quad\textrm{and}\quad r(x,t)=x-t.\end{align*} $$

Notice that $d(\mathfrak {G}_\Lambda )=r(\mathfrak {G}_\Lambda )=\Xi _{\Lambda }$ . In this context, the set $\Xi _{\Lambda }$ is called the unit space of $\mathfrak {G}_\Lambda $ .

We say that a subset E of the unit space is invariant by the groupoid $\mathfrak {G}$ if ${E=r(d^{-1}(E))}$ . We recall the following definition from [Reference RenaultRen80].

The following result relates the minimality of $(\Omega _\Lambda ,\mathbb {R}^d)$ to the minimality of the transverse groupoid.

Proof. First, observe that for every subset E of $\Xi _\Lambda $ we have

(2.3) $$ \begin{align} r(d^{-1}(E))=\{x-t\in\Xi_{\Lambda}\mid x\in E,t\in x\}. \end{align} $$

Assume that the dynamical system $(\Omega _\Lambda ,\mathbb {R}^d)$ is minimal. Suppose, by contradiction, that $E\subseteq \Xi _\Lambda $ is invariant by the groupoid $\mathfrak {G}_\Lambda $ . Define

We have that $\widehat {E}$ is open in $\Omega _\Lambda $ and by (2.3) it is invariant for the $\mathbb {R}^d$ -action on $\Omega _\Lambda $ . Then the complement of $\widehat {E}$ is an invariant non-empty closed set strictly contained in $\Omega _\Lambda $ , which contradicts the minimality of $(\Omega _\Lambda ,\mathbb {R}^d)$ .

Conversely, suppose that $(\Omega _\Lambda ,\mathbb {R}^d)$ is not minimal. Let $C\subseteq \Omega _\Lambda $ be an invariant non-empty closed set strictly contained in $\Omega _\Lambda $ . Put $E=C^c\cap \Xi _\Lambda $ . By (2.3), we have

$$ \begin{align*}E\subseteq r(d^{-1}(E)).\end{align*} $$

Since C is an invariant $\mathbb {R}^d$ -action, we get $r(d^{-1}(E))=E$ . So, E is a non-empty open set strictly contained in $\Xi _\Lambda $ invariant by the groupoid, and thus $\mathfrak {G}_\Lambda $ is not minimal.

2.4 Cut and project scheme and inter-model sets

A cut and project scheme over $\mathbb {R}^d$ is the data $(H,L)$ of a locally compact $\sigma $ -compact abelian group H, and a discrete set $L\subseteq \mathbb {R}^d\times H$ with compact quotient $( \mathbb {R}^d\times H)/L$ whose first coordinate projection on $\mathbb {R}^d$ is one-to-one and whose second coordinate projection on H is dense. A compact subset W of H that is the closure of its interior is called a window for the CPS. In the CPS the space $\mathbb {R}^d$ is called the physical space, the locally compact abelian group H is called the internal space and the set L the lattice. Following [Reference AujogueAuj16b], a CPS can also be described as a triple $(H,\Gamma ,s_H)$ where H is a locally compact $\sigma $ -compact abelian group, $\Gamma $ a countable subgroup of $\mathbb {R}^d$ and $s_{H}:\Gamma \to H$ a group homomorphism with range $s_{H}(\Gamma )$ dense in H such that the graph

is a lattice, that is, a discrete and cocompact set. When H is a Euclidean space $\mathbb {R}^n$ , for some positive integer n, we say that $(H,\Gamma ,s_{H})$ is an Euclidean CPS.

Let $(H,\Gamma ,s_{H})$ be a CPS with window W. For every w in H, the projection on $\mathbb {R}^d$ of the set $\mathcal {G}(s_H)\cap (\mathbb {R}^d\times (w+W))$ is called a model set. More generally, for every subset V of H and every w in H denote by $\curlywedge (w+V)$ the set

Definition 2.4. Let $(H,\Gamma ,s_{H})$ be a CPS over $\mathbb {R}^d$ with window W. A Delone set $\Lambda \subseteq \mathbb {R}^d$ is called an inter-model set if there exist $t\in \mathbb {R}^d$ and $w\in H$ such that

$$ \begin{align*}\curlywedge(w+\mathrm{int}(W))-t\subseteq\Lambda\subseteq \curlywedge(w+W)-t.\end{align*} $$

We say that an inter-model set $\Lambda $ is non-singular or generic if there is $(t,w)$ in $\mathbb {R}^d\times H$ such that

$$ \begin{align*}\curlywedge(w+\mathrm{int}(W))-t = \Lambda = \curlywedge(w+W)-t.\end{align*} $$

Observe that this is equivalent to the fact that the boundary of $w + W$ does not intersect the projection of $\mathcal {G}(s_H)$ in H. Additionally, if the boundary of $w + W$ has zero Haar measure we say the inter-model set is regular.

Remark 2.5. Notice that by Baire’s theorem, the fact that $\partial W$ has empty interior and $s_H(\Gamma )$ is countable, the set

$$ \begin{align*} NS:=H \setminus \bigcup_{\gamma^*\in s_H(\Gamma)} \gamma^*-\partial W \end{align*} $$

is a dense $G_\delta $ -set in H. Moreover, for every w in H, the boundary of $w + W$ does not intersect the projection of $\mathcal {G}(s_H)$ in H if and only if $w\in NS$ . In particular, for every $(t,w)$ in $\mathbb {R}^d\times H$ , the set $\curlywedge (w+W)-t$ is a non-singular inter-model set if and only if $w\in NS$ .

We say that $W'$ is irredundant if the equation $W'+w=W'$ holds only for $w=0$ in H.

The following two results are well known in the theory of model sets and will be used in the proof of Theorem A.

The first part of Proposition 2.6 follows from [Reference RobinsonRob07, Proposition 5.18, Corollary 5.10] and [Reference Lee and MoodyLM06, Proposition 4.4]. The part that assumes that the window is irredundant follows from [Reference RobinsonRob07, Theorem 5.19] and the idea of the proof of [Reference Lee and MoodyLM06, Proposition 4.6].

Let $(H,\Gamma ,s_{H})$ be a CPS in $\mathbb {R}^d$ and consider the set

with an action of $\mathbb {R}^d$ given by translation on the first coordinate. More precisely, for every $s\in \mathbb {R}^d$ and every $[(t,w)]\in \mathbb {T}_{\mathcal {G}}$ the action of s on $[(t,w)]$ is

Theorem 2.2. Let $(\mathfrak {H},\mathfrak {L},s_{\mathfrak {H}})$ be a CPS over $\mathbb {R}^d$ , let W be an irrendundant window, and let $\Omega _{\mathrm {MS}}$ be the hull of the repetitive inter-model sets generated by $(\mathfrak {H},\mathfrak {L},s_{\mathfrak {H}})$ and W. Then every point in $\Omega _{\mathrm {MS}}$ is an inter-model set, and there exists a factor map $\pi :\Omega _{\mathrm {MS}}\to \mathbb {T}_{\mathcal {G}}$ such that for every $\Lambda '$ in $\Omega _{\mathrm {MS}}$ there is $(t,w)$ in $\mathbb {R}^d\times H$ such that $\pi (\Lambda ')=[(t,w)]$ if and only if

(2.4) $$ \begin{align} \curlywedge(w+\mathrm{int}(W)) - t \subseteq\Lambda'\subseteq \curlywedge(w+W) - t. \end{align} $$

Moreover, the map $\pi $ is injective precisely on the subset of non-singular inter-model sets in $\Omega _{\mathrm {MS}}$ and the dynamical system $(\mathbb {T}_{\mathcal {G}},\mathbb {R}^d)$ is the maximal equicontinuous factor of $(\Omega _{\mathrm {MS}},\mathbb {R}^d)$ .

The proof of Theorem 2.2 is mainly in [Reference SchlottmannSch00]. The proof that $(\mathbb {T}_{\mathcal {G}},\mathbb {R}^d)$ is the maximal equicontinuous factor of $(\Omega _{\mathrm {MS}},\mathbb {R}^d)$ follows from the fact that $(\mathbb {T}_{\mathcal {G}},\mathbb {R}^d)$ is an equicontinuous factor and from the existence of points where $\pi $ is injective; see, for instance, [Reference Aujogue, Barge, Kellendonk and LenzABKL15, Lemma 3.11].

2.5 Torus parametrization

Let X be a compact space and let $(X,\mathbb {R}^d)$ be a topological dynamical system. Consider a compact abelian group $\mathbb {K}$ with a minimal action of $\mathbb {R}^d$ coming from group multiplication via a group homomorphism from $\mathbb {R}^d$ into $\mathbb {K}$ . A torus parametrization is a factor map $\pi : (X,\mathbb {R}^d)\to (\mathbb {K},\mathbb {R}^d)$ . A section of $\pi $ is a map ${s:\mathbb {K} \to X}$ such that $\pi \circ s$ is the identity on $\mathbb {K}$ . A point $x\in X$ is called singular if the fiber $\pi ^{-1}(\pi (x))$ contains more than one element. Otherwise, $x\in X$ is called non-singular. The set of non-singular points of X for $\pi $ is denoted by $R_\pi (X)$ . The following proposition was proved in [Reference Baake, Lenz and MoodyBLM07].

3.1 Defining a cocycle on the groupoid

Let $\Lambda \subseteq \mathbb {R}^d$ be a repetitive Meyer set. Let $\mathcal {B}=\{v_1,\ldots ,v_s\}\subseteq \mathbb {R}^d$ be a basis for $\langle \Lambda -\Lambda \rangle $ and let $\varphi :\langle \Lambda -\Lambda \rangle \to \mathbb {Z}^d$ be the coordinate map with respect to the basis $\mathcal {B}$ . Recall that by the repetitivity of $\Lambda $ for every $x\in \Xi _\Lambda $ we have that $\langle x-x \rangle =\langle \Lambda -\Lambda \rangle $ , and thus the address map of x associated to $\mathcal {B}$ is equal to $\varphi $ . Note that for all t and $t'$ in $\langle \Lambda - \Lambda \rangle $ we have

(3.1) $$ \begin{align} \varphi(t+t')=\varphi(t)+\varphi(t'). \end{align} $$

From Theorem 2.1, for every $x\in \Xi _{\Lambda }$ there is a unique linear map $\ell _x:\mathbb {R}^d\to \mathbb {R}^s$ such that

(3.2)

Definition 3.1. Let H be an abelian group. A cocycle on the topological groupoid $\mathfrak {G}_\Lambda $ with values in H is a map $c:\mathfrak {G}_\Lambda \to H$ such that for all composable pairs $(x,t)$ and $(z,s)$ in $\mathfrak {G}_\Lambda $ one has

$$ \begin{align*}c((x,t)\cdot(z,s))=c((x,t))+c((z,s)).\end{align*} $$

We define the maps $\Phi :\mathfrak {G}_\Lambda \to \mathbb {Z}^s$ and $L:\mathfrak {G}_\Lambda \to \mathbb {R}^s$ as follows: for every $(x,t)\in \mathfrak {G}_\Lambda $ ,

The aim of this subsection is to show that $L-\Phi $ defines a continuous cocycle on $\mathfrak {G}_\Lambda $ . For this, we first prove that L does not depend on the first coordinate. The proof of the continuity is at the end of the subsection.

The proof of this proposition is given at the end of this subsection after some lemmas.

Lemma 3.3. Let $\Lambda '$ be a relatively dense set in $\mathbb {R}^d$ . The set $\{{t}/{\|t\|_{d}}\mid t\in \Lambda '\}$ is dense in the boundary of the Euclidean unitary ball centered on the origin. In particular, for all linear maps $T:\mathbb {R}^d\to \mathbb {R}^s$ we have that

$$ \begin{align*}\|T\|_{\mathrm{op}} = \sup_{t\in x} \bigg\|T\bigg(\frac{t}{\|t\|_{d}}\bigg)\bigg\|_{s},\end{align*} $$

where $\|\cdot \|_{\mathrm {op}}$ is the operator norm.

Proof. Fix $(x,t)$ in $\mathfrak {G}_\Lambda $ . Let $u\in \mathbb {R}^d$ be such that $u\in x-t$ . In particular, $t+u\in x$ . By (3.2), we have

$$ \begin{align*}\|\varphi(u)-\ell_{x-t}(u)\|_{s}\le \xi_{x-t}\quad\textrm{and}\quad\|\varphi(u)-\ell_{x}(t+u)\|_{s}\le \xi_{x}.\end{align*} $$

Using these inequalities and (3.1), we get

$$ \begin{align*} \|\ell_x(t+u)-\ell_{x-t}(u)\|_{s}&\le\|\varphi(t+u)-\ell_x(t+u)\|_{s}+\|\varphi(t+u)-\ell_{x-t}(u)\|_{s}\\ &\le \xi_x+\|\varphi(t)+\varphi(u)-\ell_{x-t}(u)\|_{s}\\ &\le \xi_x+\|\varphi(t)\|_{s}+\xi_{x-t}. \end{align*} $$

Dividing both sides of this last inequality by

, we obtain

Taking the limit as

, we have

This, together with Lemma 3.3, implies that

, and thus concludes the proof of the lemma.

Proof of Proposition 3.2

Fix y in $\Xi _\Lambda $ . We prove that for every x in $\Xi _\Lambda $ we have $\ell _x=\ell _y$ . By (3.1), (3.2) and Lemma 3.4, for $t'$ in y we have

(3.3) $$ \begin{align} \xi_{y-t'}& =\sup_{t\in y-t'}\|\varphi(t)-\ell_{y-t'}(t)\|_s\nonumber\\ &= \sup_{t\in y-t'}\|\varphi(t)-\ell_{y}(t)\|_s\nonumber\\ &= \sup_{t+t'\in y}\|\varphi(t+t')-\varphi(t')-\ell_{y}(t)\|_s\\&= \sup_{t+t'\in y}\|\varphi(t+t')-\varphi(t')-\ell_{y}(t+t'-t')\|_s\nonumber\\ &= \sup_{t+t'\in y}\|\varphi(t+t')-\varphi(t')-\ell_{y}(t+t')+\ell_{y}(t')\|_s\nonumber\\ &\le 2\xi_y.\nonumber \end{align} $$

Fix x in $\Xi _\Lambda $ . By minimality, there is a sequence $(t_n)_{n\in \mathbb {N}}$ in $\mathbb {R}^d$ such that $y-t_n$ converges to x in $\Xi _\Lambda $ . Fix $t\in x$ and consider $\epsilon>0$ such that . There is $N\in \mathbb {N}$ such that for all $n>N$ we have

$$ \begin{align*}(y-t_n)\cap \overline{B\bigg(0,\frac{1}{\epsilon}\bigg)}=x\cap \overline{B\bigg(0,\frac{1}{\epsilon}\bigg)}.\end{align*} $$

In particular, for all $n>N$ we get $t\in y-t_n$ . Then, using Lemma 3.4 and (3.3), for every t in x we have

$$ \begin{align*}\|\varphi(t)-\ell_{y}(t)\|_s=\|\varphi(t)-\ell_{y-t_n}(t)\|_s\le 2\xi_y.\end{align*} $$

By uniqueness of the map $\ell _x$ , we conclude the proof of the proposition.

Proof. By (3.1) and Proposition 3.2, we have that $L-\Phi $ is a cocycle. Now we prove the continuity of $L-\Phi $ . Consider a sequence $\{(x_n,t_n)\}_{n\in \mathbb {N}}$ in $\mathfrak {G}_\Lambda $ that converges to $(x,t)$ in $\mathfrak {G}_\Lambda $ . By definition of convergence in the groupoid, we have that $\{x_n\}_{n\in \mathbb {N}}\subseteq \Xi _\Lambda $ converges to $x\in \Xi _\Lambda $ , and $\{t_n\}_{n\in \mathbb {N}}$ converges to t in $\mathbb {R}^d$ . Let $\epsilon $ be a positive real number less than the uniformly discrete radius of $\Lambda $ such that . There is a positive integer N such that for all $n\ge N$ we have

(3.4)

By definition of the groupoid $\mathfrak {G}_\Lambda $ , for all n in $\mathbb {N}$ we have that $t_n\in x_n$ , and also $t\in x$ . By (3.4), for every $n\ge N$ we get $t_n=t$ . Then, for every $n\ge N$ , we have

$$ \begin{align*}L(t_n)=\ell(t)\quad\text{and}\quad\Phi(x_n,t_n)= \varphi(t_n)= \varphi(t)=\Phi(x,t),\end{align*} $$

which implies the continuity of $L-\Phi $ .

3.2 Proof of Proposition 1.1

We use the following version of the Gottschalk–Hedlund theorem, due to Jean Renault, to find continuous eigenvalues of $\mathfrak {G}_\Lambda $ . This version is adapted to our context from [Reference RenaultRen80, Theorem 1.4.10] and appears in [Reference RenaultRen12].

Proof of Proposition 1.1

Let $\Lambda \subseteq \mathbb {R}^d$ be a repetitive Meyer set. Let $\mathcal {B}=\{v_1,\ldots ,v_s\}\subseteq \mathbb {R}^d$ be a basis for $\langle \Lambda -\Lambda \rangle $ and let $\varphi :\langle \Lambda -\Lambda \rangle \to \mathbb {Z}^d$ be the coordinate map with respect to the basis $\mathcal {B}$ . Let L and $\Phi $ be as in §3.1. We check that $\mathfrak {G}_\Lambda $ and the cocycle $L-\Phi :\mathfrak {G}_\Lambda \to \mathbb {R}^s$ verify the hypotheses of Theorem 3.1. By Proposition 2.3, the groupoid is minimal. By Lemma 3.5, the map $L-\Phi $ is a continuous cocycle. Let $\ell $ be the linear map given by Proposition 3.2. By (3.2), for every $x\in \Xi _{\Lambda }$ the set

$$ \begin{align*} (L-\Phi)(d^{-1}(x)) = \{\ell(t)-\varphi(t) \mid t\in x\} \end{align*} $$

is bounded. By Theorem 3.1, there is a continuous map $F:\Xi _{\Lambda }\to \mathbb {R}^s$ such that for every $(x,t)$ in $\mathfrak {G}_\Lambda $ we have

(3.5) $$ \begin{align} \ell(t)-\varphi(t) = L(x,t)-\Phi(x,t)=F\circ r(x,t)-F\circ d(x,t) = F(x-t)-F(x).\notag\\ \end{align} $$

Since F is continuous and the space $\Xi _\Lambda $ is compact there is a constant $C>0$ such that the inequality in the first part of Proposition 1.1 holds.

Now we check that $\ell $ is injective. By contradiction suppose that the kernel of $\ell $ has dimension greater than $1$ . Hence, there is an infinite subset of $\Lambda $ such that the address map is bounded on this infinite set, which gives a contradiction.

Finally, we construct the address system. Denote by $\mathbb {T}^s$ the torus $\mathbb {R}^s/\mathbb {Z}^s$ . Since $\ell $ is linear the following map defines an equicontinuous action of $\mathbb {R}^d$ on $\mathbb {T}^s$ :

$$ \begin{align*}(w,t)\in\mathbb{T}^s\times\mathbb{R}^d\longmapsto w+[\ell(t)]_{\mathbb{Z}^s}.\end{align*} $$

Now we define $\pi _{\mathrm {Ad}}:\Omega _{\Lambda }\to \mathbb {T}^s$ as follows. For every $y\in \Omega _{\Lambda }$ there exist $x\in \Xi _{\Lambda }$ and $t\in \mathbb {R}^d$ such that $y=x-t$ . Put

We verify that $\pi _{\mathrm {Ad}}$ is well defined. Indeed, suppose that for $y\in \Omega _{\Lambda }$ there are $x_1,x_2\in \Xi _{\Lambda }$ and $t_1,t_2\in \mathbb {R}^d$ such that $y=x_1-t_1=x_2-t_2$ . Thus, $x_1=x_2-(t_2-t_1)$ , and by (3.5) we have that

$$ \begin{align*} F(x_1)=F(x_2)+\ell(t_2-t_1)-\varphi(t_2-t_1), \end{align*} $$

which is equivalent to

$$ \begin{align*} F(x_1)+\ell(t_1)=F(x_2)+\ell(t_2)-\varphi(t_2-t_1). \end{align*} $$

Together with the fact that $\varphi (t_2-t_1)\in \mathbb {Z}^s$ , this implies that $\pi _{\mathrm {Ad}}$ is well defined. Now we prove the continuity of $\pi _{\mathrm {Ad}}$ . Fix $y\in \Omega _\Lambda $ and suppose that $y=x-t$ for some $x\in \Xi _\Lambda $ and $t\in \mathbb {R}^d$ . For every $y'$ close to y there is $x'$ in $\Xi _\Lambda $ close to x and there is $t'$ close to t such that $y'=x'-t'$ . By the continuity of F and $\ell $ , the map $\widetilde {\pi }_{\mathrm {Ad}}$ defined in a sufficiently small neighborhood of y by $\widetilde {\pi }_{\mathrm {Ad}}(y')=F(x')+\ell (t')$ is continuous. By the continuity of the canonical projection of $\mathbb {R}^s$ onto $\mathbb {T}^s$ we conclude that $\pi _{\mathrm {Ad}}$ is continuous at y. It remains to check that for every y in $\Omega _\Lambda $ and every t in $\mathbb {R}^d$ we have $\pi _{\mathrm {Ad}}(y-t) = \pi _{\mathrm {Ad}}(y) + [\ell (t)]_{\mathbb {Z}^s}$ . Fix y in $\Omega _\Lambda $ and fix t in $\mathbb {R}^d$ . There are $x_1$ and $x_2$ in $\Xi _\Lambda $ and $t_1$ and $t_2$ in $\mathbb {R}^d$ such that $y=x_1-t_1$ and $y-t=x_2-t_2$ . Then $x_2 = x_1-(t_1-t_2+t)$ . Using this, (3.5) and the fact that $\varphi (t_1-t_2+t)\in \mathbb {Z}^s$ , we get that

$$ \begin{align*} \pi_{\mathrm{Ad}}(y-t) &= [F(x_2)]_{\mathbb{Z}^s}+[\ell(t_2)]_{\mathbb{Z}^s} \\ &= [F(x_1) + \ell(t_1-t_2+t) -\varphi(t_1-t_2+t) ]_{\mathbb{Z}^s}+[\ell(t_2)]_{\mathbb{Z}^s} \\ &= [F(x_1) + \ell(t_1) ]_{\mathbb{Z}^s}+[\ell(t)]_{\mathbb{Z}^s} = \pi_{\mathrm{Ad}}(y) + [\ell(t)]_{\mathbb{Z}^s}, \end{align*} $$

which concludes the proof of the proposition.

4.1 Necessary condition

Let $\Lambda $ be an inter-model set for a Euclidean CPS over $\mathbb {R}^d$ with internal space $\mathbb {R}^n$ , lattice L and window W. Denote by $\Omega _{\mathrm {MS}}$ the hull of the non-singular model sets generated by these data. Repetitivity of $\Lambda $ and Proposition 2.6 imply that $\Omega _{\mathrm {MS}}=\Omega _\Lambda $ . By [Reference AujogueAuj16a, Theorem 8.1], the associated dynamical system $(\Omega _{\mathrm {MS}},\mathbb {R}^d)$ is almost automorphic (see also [Reference SchlottmannSch00, Reference Forrest, Hunton and KellendonkFHK02]). The remaining part of the proof of the necessary condition follows directly from the following proposition.

Proof. Denote by $p_1$ and by $p_2$ the orthogonal projections from $\mathbb {R}^d\times \mathbb {R}^n$ onto $\mathbb {R}^d$ and $\mathbb {R}^n$ , respectively, and put . Fix $\Lambda $ in $\Omega _{\mathrm {MS}}$ . By [Reference MoodyMoo97, Proposition 2.6(ii)], for every w in $\mathbb {R}^n$ we have that

$$ \begin{align*} \langle \curlywedge(w+W) \rangle = \Gamma. \end{align*} $$

In particular, $\langle \curlywedge (w+W) - \curlywedge (w+W) \rangle = \Gamma $ . By Proposition 2.6, there is w in $NS$ such that $\curlywedge (w+W)$ is in $\Omega _{\mathrm {MS}}$ , and thus by repetitivity

$$ \begin{align*} \langle\Lambda-\Lambda\rangle = \langle \curlywedge(w+W) - \curlywedge(w+W) \rangle = \Gamma. \end{align*} $$

We now prove that the maximal equicontinuous factor of $(\Omega _{\mathrm {MS}},\mathbb {R}^d)$ is topologically conjugate to the address system of $\Lambda $ . Fix a basis $\mathcal {B}=\{\widetilde {v}_1, \ldots , \widetilde {v}_s\}$ of L. Let $\ell $ be the linear map given by the Proposition 1.1 applied to $\Lambda $ with the basis $p_1(\mathcal {B})$ for $\Gamma $ and let $(\mathbb {T}^s,\mathbb {R}^d)$ be the corresponding address system. Denote by $\psi :\mathbb {R}^s\to \mathbb {R}^d\times \mathbb {R}^n$ the linear isomorphism sending the canonical basis of $\mathbb {R}^s$ onto $\{\widetilde {v}_1, \ldots , \widetilde {v}_s\}$ , that is,

$$ \begin{align*}\psi(u_1,\ldots,u_s)=u_{1}\widetilde{v}_1+\cdots+u_{s}\widetilde{v}_s.\end{align*} $$

By (2.2), for every $t\in \mathbb {R}^d$ we have

(4.1) $$ \begin{align} p_1(\psi(\ell(t)))=t. \end{align} $$

Define the map $\Psi :\mathbb {T}^s\to \;\mathbb {T}_{\mathcal {G}}$ by $\Psi ([w]_{\mathbb {Z}^s})=[\psi (w)]_{L}$ . Note that $\Psi $ is a homeomorphism. By (4.1), for all $t\in \mathbb {R}^d$ and $[w]\in \mathbb {T}^{s}$ , we have

$$ \begin{align*} \Psi([w]_{\mathbb{Z}^s}+[\ell(t)]_{\mathbb{Z}^s} ) &= \Psi([w+\ell(t)]_{\mathbb{Z}^s} ) \\ &= [\psi(w + \ell(t))]_L = [\psi(w)]_L +[\psi(\ell(t))]_L \\ &= [\psi(w)]_L +[(p_1(\psi(\ell(t))),p_2(\psi(\ell(t))))]_L \\ &= [\psi(w)]_L +[(t,p_2(\psi(\ell(t))))]_L. \end{align*} $$

To prove that $\Psi $ conjugates the address system with the maximal equicontinuous factor $(\mathbb {R}^d\times \mathbb {R}^n / L, \mathbb {R}^d)$ , we need to show that for every $t\in \mathbb {R}^d$ ,

$$ \begin{align*}p_2(\psi(\ell(t)))=0.\end{align*} $$

By Remark 2.5, Proposition 2.6 and the fact that the window W has non-empty interior, there is w in $NS$ such that $0\in w+W$ and the set $\curlywedge (w+W)$ is in $\Omega _{\mathrm {MS}}$ . Put . We have that $\Lambda _0$ is in $\Xi _\Lambda $ . Observe that $\varphi $ is also the address map for $\Lambda _0$ associated to the basis $p_1(\mathcal {B})$ . By Proposition 1.1, there is a constant $\widehat {C}>0$ such that for every $t\in \Lambda _0$ we have

$$ \begin{align*} \|p_2(\psi(\varphi(t)))-p_2(\psi(\ell(t)))\|_d\le\widehat{C}. \end{align*} $$

Together with the fact that $p_2(\psi (\varphi (\Lambda _0)))=p_2(s_{\mathbb {R}^n}(\Lambda _0))\subseteq w+W$ , this implies that the map $p_2\circ \psi \circ \ell $ is uniformly bounded on $\Lambda _0$ . Using that $\Lambda _0$ is relatively dense in $\mathbb {R}^d$ and that $p_2\circ \psi \circ \ell $ is linear, we get that $p_2(\psi (\ell (\mathbb {R}^d)))$ is bounded, which implies that $p_2(\psi (\ell (\mathbb {R}^d)))=0$ . We conclude that $(\mathbb {T}^s,\mathbb {R}^d)$ and $(\mathbb {T}_{\mathcal {G}},\mathbb {R}^d)$ are topologically conjugated, finishing the proof of the lemma.

4.2 Sufficient condition

4.2.1 The Lagarias cut and project scheme

Let $\Lambda $ be a repetitive Meyer set in $\mathbb {R}^d$ and suppose that $\langle \Lambda -\Lambda \rangle $ has rank $s>d$ . Let $\mathcal {B}$ be a basis of $\langle \Lambda -\Lambda \rangle $ formed by vectors $\{v_1,\ldots ,v_s\}\subseteq \mathbb {R}^d$ and let $\varphi :\langle \Lambda -\Lambda \rangle \to \mathbb {Z}^d$ be the coordinate map with respect to the basis $\mathcal {B}$ . Fix $\Lambda _0$ in $\Xi _\Lambda $ . Remember that since $0\in \Lambda _0$ , we have $\langle \Lambda -\Lambda \rangle =\langle \Lambda _0-\Lambda _0\rangle =\langle \Lambda _0\rangle $ and that $\varphi $ is also the address map for $\Lambda _0$ . Let $\ell :\mathbb {R}^d\to \mathbb {R}^s$ be the linear map given by Proposition 1.1. Define $\phi :\mathbb {R}^s\to \mathbb {R}^d$ by $\phi (u_1,\ldots ,u_s)=u_{1}v_1+\cdots +u_{s}v_s$ . By (2.2), for every $t\in \mathbb {R}^d$ we have

(4.2) $$ \begin{align} {\phi}{\circ}{\ell}(t)=t. \end{align} $$

In particular,

(4.3) $$ \begin{align} \mathrm{Ker}(\ell)=\{0\}\quad\textrm{and} \quad\mathrm{Im}(\phi)=\mathbb{R}^d. \end{align} $$

Put

and note that the dimension of $\mathrm {Ker}(\phi )$ is n. Let

be an orthonormal basis for $\mathrm {Ker}(\phi )$ . Notice that for every $1\le j\le s$ we have that the vector

belongs to $\mathrm {Ker}(\phi )$ , where $e_j$ is the jth canonical coordinate vector. For every $j\in \{1,\ldots , s\}$ denote by $(\alpha _{j,1},\ldots \alpha _{j,n})$ the coordinates of $w_j$ in the basis $\mathcal {B}'$ , and define for every $j\in \{1, \ldots , s\}$ the vectors

In the proof of [Reference LagariasLag99, Theorem 3.1], Lagarias proved that the set

is $\mathbb {Z}$ -linearly independent in $\mathbb {R}^d\times \mathbb {R}^n$ and generates a full-rank lattice. Denote by $\widetilde {L}$ the lattice generated by $\widetilde {\mathcal {B}}$ . Denote by $p_1$ and $p_2$ the orthogonal projections of $ \mathbb {R}^d\times \mathbb {R}^n$ onto $\mathbb {R}^d$ and $\mathbb {R}^n$ , respectively. By construction, $p_1$ is injective on $\widetilde {L}$ and its image is $\langle \Lambda -\Lambda \rangle $ . Denote by $\psi :\mathbb {R}^s\to \mathbb {R}^d\times \mathbb {R}^n$ the linear isomorphism sending the canonical basis of $\mathbb {R}^s$ onto $\{\widetilde {v}_1, \ldots , \widetilde {v}_s\}$ , that is,

$$ \begin{align*}\psi(u_1,\ldots,u_s)=u_{1}\widetilde{v}_1+\cdots+u_{s}\widetilde{v}_s.\end{align*} $$

In the proof of [Reference LagariasLag99, Theorem 3.1], it was proved that for every t in $\langle \Lambda -\Lambda \rangle $ we have

(4.4) $$ \begin{align} \| p_2(\psi(\varphi(t))) \|_n = \| \varphi(t) - \ell(t) \|_s. \end{align} $$

Lemma 4.2. Let $\Lambda $ be a repetitive Meyer set in $\mathbb {R}^d$ . If the address system of $\Lambda $ associated with $\mathcal {B}$ is a topological factor of $(\Omega _\Lambda ,\mathbb {R}^d)$ , then $p_2(\widetilde {L})$ is dense in $\mathbb {R}^n$ .

Proof. The proof is by contradiction. Assume that $p_2(\widetilde {L})$ is not dense. Then there is a non-empty closed ball $V\subseteq \mathbb {R}^n$ such that $p_2(\widetilde {L})\cap V=\{\emptyset \}$ . In particular,

(4.5) $$ \begin{align} \widetilde{L}\cap (\mathbb{R}^d\times V)=\{\emptyset\}. \end{align} $$

By Proposition 1.1 and (4.4), there is a constant $\widehat {C}>0$ such that for every $t\in \Lambda _0$ we have

$$ \begin{align*} \max\{\| p_2(\psi(\varphi(t))) \|_n, \|p_2(\psi(\varphi(t)))-p_2(\psi(\ell(t)))\|_n \} \le\widehat{C}. \end{align*} $$

Therefore the linear map $p_2\circ \psi \circ \ell $ is uniformly bounded on $\Lambda _0$ , which is relatively dense. Then, for all $t\in \mathbb {R}^d$ , we have

(4.6) $$ \begin{align} p_2\circ\psi\circ\ell(t)=0. \end{align} $$

Consider the dynamical system defined on the space $(\mathbb {R}^d\times \mathbb {R}^n)/\widetilde {L}$ with the following $\mathbb {R}^d$ -action: for every $ t \in \mathbb {R}^d$ and every $w\in (\mathbb {R}^d\times \mathbb {R}^n)/\widetilde {L},$

Define the map $\Psi :\mathbb {T}^s\to (\mathbb {R}^d\times \mathbb {R}^n)/\widetilde {L}$ by $\Psi ([w]_{\mathbb {Z}^s})=[\psi (w)]_{\widetilde {L}}$ for every $[w]_{\mathbb {Z}^s}$ in $(\mathbb {R}^d\times \mathbb {R}^n)/\widetilde {L}$ . By (4.6), the map $\Psi $ is a topological conjugacy between the address system of $\Lambda $ and the dynamical system just defined $((\mathbb {R}^d\times \mathbb {R}^n)/\widetilde {L}, \mathbb {R}^d)$ . Let $\pi _{\mathrm {Ad}}$ be the address homomorphism defined in Proposition 1.1. Since we are assuming that $\pi _{\mathrm {Ad}}$ is a factor of $(\Omega _\Lambda ,\mathbb {R}^d)$ , we have that the map $\Psi \circ \pi _{\mathrm {Ad}}$ is also a factor from $(\Omega _\Lambda ,\mathbb {R}^d)$ to $((\mathbb {R}^d\times \mathbb {R}^n)/\widetilde {L}, \mathbb {R}^d)$ . By the repetitivity of $\Lambda $ we have that $(\Omega _\Lambda ,\mathbb {R}^d)$ is minimal, and then the factor $((\mathbb {R}^d\times \mathbb {R}^n)/\widetilde {L}, \mathbb {R}^d)$ is also minimal. But the set $[\mathbb {R}^d\times V]_{\widetilde {L}}$ is closed and $\mathbb {R}^{d}$ -invariant, and by (4.5), it is strictly contained in $(\mathbb {R}^d\times \mathbb {R}^n)/\widetilde {L}$ , which is a contradiction to the minimality of $((\mathbb {R}^d\times \mathbb {R}^n)/\widetilde {L}, \mathbb {R}^d)$ .

Put on $\langle \Lambda - \Lambda \rangle $ . By Lemma 4.2 if the address system of $\Lambda $ is a topological factor of $(\Omega _\Lambda ,\mathbb {R}^d)$ the triple $(\mathbb {R}^n,\langle \Lambda -\Lambda \rangle ,s_L)$ is a CPS and we call it the Lagarias CPS for $\Lambda $ .

Recall that a window is irredundant if its redundancies group is trivial (see §2.4). By compactness, every window in $\mathbb {R}^n$ is irredundant. By Theorem 2.1 and (4.4), the set $\overline {s_L(\Lambda _0)}\subseteq \mathbb {R}^n$ is a compact. Together with Proposition 5.2 we obtain the following result.

Lemma 4.3. Let $\Lambda $ be a repetitive Meyer set in $\mathbb {R}^d$ and let $\langle \Lambda -\Lambda \rangle $ be the subgroup of $\mathbb {R}^d$ generated by $\Lambda -\Lambda $ . Put $n=\mathrm {rank}(\langle \Lambda -\Lambda \rangle )-d$ and assume that $n>0$ . Also assume that some address system of $\Lambda $ is a topological factor of $(\Omega _\Lambda ,\mathbb {R}^d)$ . Let $(\mathbb {R}^n,\langle \Lambda -\Lambda \rangle ,s_L)$ be the Lagarias CPS for $\Lambda $ . For every $\Lambda _0$ in $\Xi _\Lambda $ the set $\overline {s_L(\Lambda _0)}$ is an irredundant window.

From the proof of Lemma 4.2 and by Lemma 4.3 we obtain the following lemma.

Lemma 4.4. Let $\Lambda $ be a repetitive Meyer set in $\mathbb {R}^d$ and let $\langle \Lambda -\Lambda \rangle $ be the subgroup of $\mathbb {R}^d$ generated by $\Lambda -\Lambda $ . Put $n=\mathrm {rank}(\langle \Lambda -\Lambda \rangle )-d$ and assume that $n>0$ . Also assume that some address system of $\Lambda $ is a topological factor of $(\Omega _\Lambda ,\mathbb {R}^d)$ . Let $(\mathbb {R}^n,\langle \Lambda -\Lambda \rangle ,s_L)$ be the Lagarias CPS for $\Lambda $ . For every $\Lambda _0$ in $\Xi _\Lambda $ , let $\Omega _{\mathrm {MS}}$ be the hull of the generic inter-model sets generated by $(\mathbb {R}^n,\langle \Lambda -\Lambda \rangle ,s_L)$ and the window $\overline {s_L(\Lambda _0)}$ . Then the maximal equicontinuous factor of $(\Omega _{\mathrm {MS}},\mathbb {R}^d)$ is topologically conjugated to each address system of $\Lambda $ .

4.2.2 Proof of sufficient condition

The main technical step in the proof of the sufficient condition is the following lemma, proved in §5.

Main Technical Lemma. Let $\Lambda \subseteq \mathbb {R}^d$ be a repetitive Meyer set and let $\Gamma $ be the subgroup of $\mathbb {R}^d$ generated by $\Lambda $ . Let $(H',\Gamma , s_{H'})$ be a CPS and suppose that $W'=\overline {s_{H'}(\Lambda )}$ is a window. Let $\Omega _{\mathrm {MS}}$ be the hull of the generic model sets generated by $(H',\Gamma ,s_{H'})$ and $W'$ . Then there is a factor map

$$ \begin{align*} \widetilde{\pi}: \Omega_{\Lambda}\to \Omega_{\mathrm{MS},\mathrm{me}}, \end{align*} $$

such that if $(\Omega _{\Lambda },\mathbb {R}^d)$ is an almost automorphic extension of $(\Omega _{\mathrm {MS}, \mathrm {me}},\mathbb {R}^d)$ for $\widetilde {\pi }$ , then there are $\Lambda _0$ in $\Omega _{\Lambda }$ and a non-singular inter-model set $\Lambda _1$ in $\Omega _{\mathrm {MS}}$ such that $\Lambda _0=\Lambda _1$ .

Proof of sufficient condition in Theorem A

Let $\Lambda $ be a repetitive Meyer set in $\mathbb {R}^d$ and let $\langle \Lambda -\Lambda \rangle $ be the subgroup of $\mathbb {R}^d$ generated by $\Lambda -\Lambda $ . Assume that $\mathrm {rank}(\langle \Lambda -\Lambda \rangle )=s>d$ , that some address system of $\Lambda $ is a topological factor of $(\Omega _\Lambda ,\mathbb {R}^d)$ and that $(\Omega _{\Lambda },\mathbb {R}^d)$ is an almost automorphic extension of this address system. Since the address systems are topologically conjugated among them we have that every address system of $\Lambda $ is a topological factor of $(\Omega _\Lambda ,\mathbb {R}^d)$ and that $(\Omega _{\Lambda },\mathbb {R}^d)$ is an almost automorphic extension of every address system of $\Lambda $ .

Let $(\mathbb {R}^n,\langle \Lambda -\Lambda \rangle , s_{L})$ be the Lagarias CPS for $\Lambda $ where $n=s-d$ . Fix $\Lambda _*$ in $\Xi _\Lambda $ and recall that by the repetitivity of $\Lambda $ we have that $\Omega _\Lambda =\Omega _{\Lambda _*}$ . By Lemma 4.3, the set $W'=\overline {s_L(\Lambda _*)}$ is an irredundant window. Denote by $\Omega _{\mathrm {MS}}$ the hull of the generic inter-model sets generated by $(\mathbb {R}^n,\langle \Lambda -\Lambda \rangle ,s_L)$ and $W'$ . By Lemma 4.4, the maximal equicontinuous factor of $(\Omega _{\mathrm {MS}}, \mathbb {R}^d)$ is topologically conjugated to every address system of $\Lambda $ which agrees with the address systems of $\Lambda _*$ by Proposition 1.1. By hypothesis, the dynamical system $(\Omega _{\Lambda _*},\mathbb {R}^d)$ is an almost automorphic extension of every address system of $\Lambda _*$ , and then it is also an almost automorphic extension of $(\Omega _{\mathrm {MS}, \mathrm {me}}, \mathbb {R}^d)$ . By the main technical lemma applied to $\Lambda _*$ and $(\mathbb {R}^n,\langle \Lambda -\Lambda \rangle , s_L)$ , there are $\Lambda _0\in \Omega _{\Lambda _*}$ and $\Lambda _1\in \Omega _{\mathrm {MS}}$ such that $\Lambda _0=\Lambda _1$ . By the minimality of $(\Omega _{\Lambda _*},\mathbb {R}^d)$ we have that $\Omega _{\Lambda _*}$ is equal to the hull of $\Lambda _0$ which is equal to the hull of the generic model sets generated by a Euclidean CPS. Since $\Omega _\Lambda =\Omega _{\Lambda _*}$ by Proposition 2.6 we conclude that $\Lambda $ is an inter-model set generated by a CPS with Euclidean internal space, finishing the proof of the sufficient condition.

5.1 The optimal CPS and the optimal window

Let $\Lambda $ be a repetitive Meyer set in $\mathbb {R}^d$ and let $\Gamma $ be the subgroup of $\mathbb {R}^d$ generated by $\Lambda $ . Define $\Xi ^{\Gamma }$ as the collection of all $\Lambda '\in \Omega _\Lambda $ having support in $\Gamma $ :

Observe that $\Xi _{\Lambda }\subseteq \Xi ^{\Gamma }$ . We consider the combinatorial topology on $\Omega _\Lambda $ , which is obtained from the distance

$$ \begin{align*} \mathrm{dist}(\Lambda',\Lambda")=\bigg\{\frac{1}{R+1}\bigg| \Lambda' \cap \overline{B(0,R)}\ {=}\ \Lambda" \cap \overline{B(0,R)}\bigg\}. \end{align*} $$

The combinatorial topology is always strictly finer than the usual topology on $\Omega _{\Lambda }$ , and on the transversal $\Xi _\Lambda $ both topologies coincide. We endow $\Xi ^{\Gamma '}$ with the combinatorial topology. We say that $\Lambda '$ and $\Lambda "$ in $\Omega _\Lambda $ are strongly regionally proximal, denoted $\Lambda '\sim _{\mathrm {srp}}\Lambda "$ , if for each $R>0$ there are $\Lambda _1,\Lambda _2\in \Omega _\Lambda $ and $t\in \mathbb {R}^d$ such that

$$ \begin{align*} \begin{array}{c} \Lambda'\cap \overline{B(0,R)}\ {=}\ \Lambda_1\cap\overline{B(0,R)},\\ \Lambda"\cap\overline{B(0,R)}\ {=}\ \Lambda_2\cap\overline{B(0,R)},\\ (\Lambda_{1}-t)\cap\overline{B(0,R)}\ {=}\ (\Lambda_{2}-t)\cap\overline{B(0,R)}. \end{array} \end{align*} $$

Since $\Lambda $ is a repetitive Meyer set we have that the strongly regionally proximal relation is a closed $\mathbb {R}^d$ -invariant equivalent relation on $\Omega _\Lambda $ , and moreover, it agrees with the equicontinuous relation; see [Reference Barge and KellendonkBK13]. In particular, the quotient $\Omega _\Lambda /\sim _{\mathrm {srp}}$ gives the maximal equicontinuous factor.

In the following proposition we recall some results in [Reference AujogueAuj16a] which allow us to introduce the optimal CPS and optimal window for a Meyer set. More precisely, part (1) is deduced by [Reference AujogueAuj16a, Proposition 4.4 and Lemma 4.5], part (2) comes from [Reference AujogueAuj16a, Proposition 6.1 and Definition 6.2], and part (3) is in [Reference AujogueAuj16a, Theorem 7.1].

We remark that Aujogue defined $s_H$ in [Reference AujogueAuj16a] with a plus sign instead of a minus as we do. So some results that we use from [Reference AujogueAuj16a, Reference AujogueAuj16b] look slightly different since we need to make a sign correction. From Proposition 5.1, the triple $(H,\Gamma ,s_{H})$ is a CPS. Moreover, by [Reference AujogueAuj16a, Theorem 6.3], the set $[\Xi _{\Lambda }]_{\mathrm {srp}}$ is a window for $(H,\Gamma ,s_{H})$ . The CPS $(H,\Gamma ,s_{H})$ and the window $[\Xi _{\Lambda }]_{\mathrm {srp}}$ are called the optimal CPS and the optimal window for $\Lambda $ , respectively. Indeed, in [Reference AujogueAuj16b], the author proved that the model set that it defines,

satisfies that for every model set M that includes $\Lambda $ we have $\Lambda \subseteq \underline {\Lambda }\subseteq M$ .

Finally, we recall some results in [Reference AujogueAuj16b] that we use in the proof of the main technical lemma $^{\prime }$ . The first result allows us to prove that a compact and irredundant set is a window.

Proposition 5.2. [Reference AujogueAuj16b, Proposition 3.3]

Let $\Lambda $ be a repetitive Meyer set in $\mathbb {R}^d$ and let $\Gamma $ the subgroup of $\mathbb {R}^d$ generated by $\Lambda $ . Let $(H,\Gamma , s_{H})$ and W be the optimal CPS and window for $\Lambda $ , respectively. Suppose that $(H',\Gamma ,s_{H'})$ is a CPS such that the closure $W'$ of the set $s_{H'}(\Lambda )$ is compact and irredundant in $H'$ . Then there is a continuous open and onto morphism

$$ \begin{align*} \theta: H\to H' \end{align*} $$

such that $s_{H'} \kern1.2pt{=}\kern1.5pt \theta \circ s_H$ on $\Gamma .$ Moreover, the set $W'$ is a window in $H'$ and $W'\kern1.2pt{=}\kern1.5pt\theta ([\Xi _{\Lambda }]_{\mathrm {srp}})$ .

In the following result we recall the definition of a map that we use to construct the maps $\pi _1$ and $\widetilde {\pi }$ in the statement of the main technical lemma $^{\prime }$ .

Lemma 5.3. [Reference AujogueAuj16b, Lemmas 3.4, 3.5, 3.6]

Let $\Lambda $ be a repetitive Meyer set in $\mathbb {R}^d$ and let $\Gamma $ be the subgroup of $\mathbb {R}^d$ generated by $\Lambda $ . Suppose that $(H',\Gamma ,s_{H'})$ is a CPS such that the closure $W'$ of the set $s_{H'}(\Lambda )$ is compact and irredundant in $H'$ . We have that each $\Lambda '$ in $\Xi ^\Gamma $ defines a unique element $w_\Lambda '$ through

$$ \begin{align*} \{w_\Lambda'\} = \bigcap_{\gamma\in \Lambda'} s_{H'}(\gamma)-W'. \end{align*} $$

Define the map

$$ \begin{align*}\begin{array}{ll} \omega: &\hspace{-9pt} \Xi^\Gamma\to H'\\ &\hspace{-9pt} \Lambda'\mapsto w_{\Lambda'}. \end{array}\end{align*} $$

We have that $\omega $ is uniformly continuous for the combinatorial topology, and for all $\Lambda '\in \Xi ^\Gamma $ and $\gamma \in \Gamma $ we have:

1. (1) $\omega (\Lambda '-\gamma )=\omega (\Lambda ')-s_{H'}(\gamma )$ ;

2. (2) $\omega (\Lambda ')=-\theta ([\Lambda ']_{\mathrm {srp}})$ , where $\theta $ is the morphism in Proposition 5.2.

5.2 Proof of main technical lemma $^{\prime }$

Let $\Lambda \subseteq \mathbb {R}^d$ be a repetitive Meyer set and let $\Gamma $ be the subgroup of $\mathbb {R}^d$ generated by $\Lambda $ . Let $(H',\Gamma ,s_{H'})$ be a CPS and assume that $W'=\overline {s_{H'}(\Lambda )}$ is a compact and irredundant window in $H'$ . Let $\Omega _{\mathrm {MS}}$ be the hull of inter-model sets generated by $(H',\Gamma ,s_{H'})$ and $W'$ . Recall that the maximal equicontinuous factor $\Omega _{\mathrm {MS}, \mathrm {me}}$ can be obtained by the quotient $(\mathbb {R}^d\times H')/ \mathcal {G}(s_{H'})$ and denote by $\pi _0$ be the maximal equicontinuous factor map from $\Omega _{\mathrm {MS}}$ to $\Omega _{\mathrm {MS}, \mathrm {me}}$ .

5.2.1 Construction of $\widetilde {\pi }$

We now construct the map $\widetilde {\pi }: \Omega _\Lambda \to \Omega _{\mathrm {MS}, \mathrm {me}}$ . For every $(t,w)$ in $\mathbb {R}^d\times H'$ we denote by $[(t,w)]$ its equivalent class in $\Omega _{\mathrm {MS}, \mathrm {me}}$ . For every $\widetilde {\Lambda }$ in $\Omega _{\Lambda }$ there is $t\in \mathbb {R}^d$ such that $\widetilde {\Lambda }-t$ is in $\Xi ^\Gamma $ , and we define $\widetilde {\pi }(\widetilde {\Lambda })$ by

We verify that $\widetilde {\pi }$ is well defined. Assume that there is s in $\mathbb {R}^d$ such that $\widetilde {\Lambda }-s$ is in $\Xi ^\Gamma $ . Observe that $t-s$ is in $\Gamma $ . By part (1) in Lemma 5.3, we have that

$$ \begin{align*} (-t,\omega(\widetilde{\Lambda}-t)) &= (-t+s-s,\omega(\widetilde{\Lambda}-(t+s-s)))\\ &= (-s - (t-s),\omega(\widetilde{\Lambda}-s)-s_{H'}(t-s)) \\ &= (-s,\omega(\widetilde{\Lambda}-s)) - ( t-s,s_{H'}(t-s)). \end{align*} $$

Since $( t-s,s_{H'}(t-s))$ belongs to $\mathcal {G}(s_{H'})$ , we have that

$$ \begin{align*}[(-t,\omega(\widetilde{\Lambda}-t))] = [(-s,\omega(\widetilde{\Lambda}-s))],\end{align*} $$

and hence $\widetilde {\pi }$ is well defined.

We now check that $\widetilde {\pi }$ commutes with the $\mathbb {R}^d$ action on $\Omega _{\Lambda }$ and on $\Omega _{\mathrm {MS}, \mathrm {me}}.$ Let $\widetilde {\Lambda }$ be in $\Omega _{\Lambda }$ and t be in $\mathbb {R}$ . There are s and $s'$ in $\mathbb {R}^d$ such that $\widetilde {\Lambda }-s$ and $(\widetilde {\Lambda }-t)-s'= \widetilde {\Lambda }-(t+s')$ are in $\Xi ^\Gamma $ . Notice that $t+s'-s$ belongs to $\Gamma $ . Again, by part (1) in Lemma 5.3 we have

$$ \begin{align*} (-s',\omega((\widetilde{\Lambda}-t)-s')) &= (-s',\omega((\widetilde{\Lambda}-s)-(t+s'-s)))\\ &= (-s',\omega(\widetilde{\Lambda}-s)-s_{H'}(t+s'-s))\\ &= (-s'+(t+s'-s),\omega(\widetilde{\Lambda}-s))-(t+s'-s, s_{H'}(t+s'-s))\\ &= (t-s,\omega(\widetilde{\Lambda}-s))-(t+s'-s, s_{H'}(t+s'-s)). \end{align*} $$

Since $(t+s'-s, s_{H'}(t+s'-s))$ is in $\mathcal {G}(s_{H'})$ we have

$$ \begin{align*} \widetilde{\pi}(\widetilde{\Lambda}-t) = [(-s',\omega((\widetilde{\Lambda}-t)-s'))] = [(-s,\omega(\widetilde{\Lambda}-s))]+[(t,0)] = \widetilde{\pi}(\widetilde{\Lambda}) + [(t,0)].\end{align*} $$

Now we prove that $\widetilde {\pi }$ is continuous. Let $\Lambda '$ be $\Omega _{\mathrm {MS}}$ and let U be a neighborhood of $0$ in $\Omega _{\mathrm {MS}, \mathrm {me}}$ . We can assume that $U=[B(0,r_0)\times U_{H'}]$ where $r_0>0$ and $U_{H'}$ is a neighborhood of $0$ in $H'$ . There exists $t'\in \Lambda '$ such that $\Lambda '-t'\in \Xi _{\Lambda }\subseteq \Xi ^\Gamma $ . For $r>0$ , denote

$$ \begin{align*}C_r=\bigcap_{\gamma\in (\Lambda'-t')\cap\overline{B(0,r)}}s_{H'}(\gamma)-W',\end{align*} $$

and observe that for $r>r'$ we have $C_r\subseteq C_{r'}$ . By Lemma 5.3,

(5.2) $$ \begin{align} \bigcap_{r>0}C_r=\{\omega(\Lambda'-t')\}. \end{align} $$

Now we prove that there is $r'>0$ such that for every $r\ge r'$ ,

(5.3) $$ \begin{align} C_r \subseteq \omega(\Lambda'-t')+U_{H'}. \end{align} $$

By contradiction, suppose that there is an increasing sequence $(r_i)_{i\in \mathbb {N}}$ of positive real numbers converging to infinity as i goes to infinity, such that ${(C_{r_i}-\omega (\Lambda '-t'))\cap U_{H'}^c\neq \emptyset }$ . Then, for every $i\in \mathbb {N}$ , there is

$$ \begin{align*} x_{i}\in (C_{r_i}-\omega(\Lambda'-t'))\cap U_{H'}^c \end{align*} $$

since for every $i,j$ in $\mathbb {N}$ with $j\ge i$ we have $C_{r_j}\subseteq C_{r_i}$ . By compactness of $C_{r_1}$ there is an accumulation point $\tilde {x}$ of $(x_{i})_{i\in \mathbb {N}}$ in $U_{H'}^c$ , and thus $\tilde {x}\neq 0$ . But $\tilde {x}$ also belongs to $\bigcap _{r>0} C_r -\omega (\Lambda '-t')$ which is $\{0\}$ by (5.2), giving the desired contradiction.

Put

and consider the set

For every $\varepsilon>0$ sufficiently small the set

is an open neighborhood of $\Lambda '$ . Fix $\varepsilon < r_0$ . By the definition of R, for every $\Lambda " $ in $V_\varepsilon $ there are t in $B(0,\varepsilon )$ and $\widetilde {\Lambda }$ in T such that

$$ \begin{align*} (\Lambda"-(t'-t)) \cap \overline{B(0,r')} = (\widetilde{\Lambda}-t' )\cap \overline{B(0,r')} = (\Lambda'-t' )\cap \overline{B(0,r')}. \end{align*} $$

Put . We have and since $\Lambda '-t'$ is in $\Xi _\Lambda $ we also have that $\Lambda "-t"$ is in $\Xi _\Lambda \subseteq \Xi ^{\Gamma }$ . Then

$$ \begin{align*}\bigcap_{\gamma\in (\Lambda"-t")\cap\overline{B(0,r')}}s_{H'}(\gamma)-W'=\bigcap_{\gamma\in (\Lambda'-t')\cap\overline{B(0,r')}}s_{H'}(\gamma)-W'.\end{align*} $$

Together with (5.3), this implies $\omega (\Lambda "-t")\in \omega (\Lambda '-t')+U_{H'}$ . Therefore, $\widetilde {\pi }(\Lambda ")=[-t",\omega (\Lambda "-t")]$ is included in

$$ \begin{align*} [-t'+(t'-t"),\omega(\Lambda'-t')+U_{H'}]&\subseteq[-t'+B(0,\delta),\omega(\Lambda'-t')+U_{H'}]\\ &=[-t',\omega(\Lambda'-t')]+[B(0,\delta),U_{H'}], \end{align*} $$

showing the continuity of $\widetilde {\pi }$ at $\Lambda '$ in $\Omega _{\mathrm {MS}}$ .

Finally, since the $\mathbb {R}^d$ -action on $\Omega _{\mathrm {MS}, \mathrm {me}}$ is minimal we have that $\widetilde {\pi }$ is surjective, which concludes the proof that $\widetilde {\pi }$ is a factor map.

5.2.2 Definition of $\pi _1$

Recall that $R(\Omega _{\mathrm {MS}})$ denotes the set of non-singular points of $\Omega _{\mathrm {MS}}$ for $\pi _0$ as defined in §2.5. By definition, all sections of $\pi _0$ agree on $\pi _0(R(\Omega _{\mathrm {MS}}))$ . Let $\widetilde {s}:\Omega _{\mathrm {MS}, \mathrm {me}} \to \Omega _{\mathrm {MS}}$ be a section of $\pi _0$ . Put , and define the surjective map $\pi _1: \Omega _{\Lambda }^0\to R(\Omega _{\mathrm {MS}})$ by .

By the continuity of $\widetilde {\pi }$ and Proposition 2.7, the map $\pi _1$ is also continuous. Since $\widetilde {s}$ is a section of $\pi _0$ , for every $\Lambda '$ in $\Omega _{\Lambda }^0$ we have

(5.4) $$ \begin{align} \widetilde{\pi}(\Lambda') = \pi_0 \circ \pi_1(\Lambda'). \end{align} $$

Since $\widetilde {s}$ commutes with the action of $\mathbb {R}^d$ on the set $\pi _0(R(\Omega _{\mathrm {MS}}))$ , we get that for every $\Lambda '$ in $\Omega _{\Lambda }^0$ and t in $\mathbb {R}^d$ ,

$$ \begin{align*} \pi_1(\Lambda'-t) = \pi_1(\Lambda')-t. \end{align*} $$

5.2.3 Proof of (5.1)

Fix $\Lambda _1$ in $R_{\pi _0}(\Omega _{\mathrm {MS}})$ . We prove that (5.1) holds. First, we assume that $\Lambda _1$ is in $\pi _1(\Omega ^0_{\Lambda }\cap \Xi ^\Gamma )$ . By Theorem 2.2, if $\pi _0(\Lambda _1)=[(t,w)]$ then

(5.5) $$ \begin{align} \curlywedge(w+\mathrm{int}(W')) = \Lambda_1+t = \curlywedge(w+W'). \end{align} $$

Observe that, by definition of $\widetilde {\pi }$ , for every $\Lambda '$ in $\Omega ^0_{\Lambda }\cap \Xi ^\Gamma $ we have $\widetilde {\pi }(\Lambda ')=[(0,\omega (\Lambda ')]$ . In addition, if $\Lambda '$ satisfies $\pi _1(\Lambda ')=\Lambda _1$ then, using (5.4), we get

$$ \begin{align*}\pi_0(\Lambda_1)=\pi_0\circ\pi_1(\Lambda')=\widetilde{\pi}(\Lambda')=[(0,\omega(\Lambda'))].\end{align*} $$

Together with (5.5), this implies that for every $\Lambda '\in \Omega ^0_{\Lambda }\cap \Xi ^\Gamma $ such that $\pi _1(\Lambda ')=\Lambda _1$ , we have

(5.6) $$ \begin{align} \Lambda_1 = \{\gamma \in \Gamma \mid s_{H'}(\gamma) \in \omega(\Lambda') + W'\}. \end{align} $$

By Proposition 5.2 and part (2) of Lemma 5.3, we have

(5.7) $$ \begin{align} -\omega(\Xi_\Lambda)=\theta([\Xi_\Lambda]_{\mathrm{srp}})=W'. \end{align} $$

Since $\Lambda '\in \Xi ^\Gamma $ , and for every $\gamma \in \Lambda '$ we have $\Lambda '-\gamma \in \Xi _{\Lambda }$ , using part (1) of Lemma 5.3, we get $\omega (\Lambda '-\gamma )=\omega (\Lambda ')-s_{H'}(\gamma )$ . Together with (5.6) and (5.7), this implies that for every $\gamma $ in $\Lambda '$ we have

$$ \begin{align*} \omega(\Lambda'-\gamma) \in \omega(\Xi_{\Lambda}) &\Longleftrightarrow\, s_{H'}(\gamma) \in \omega(\Lambda') -\omega(\Xi_{\Lambda})\\ &\Longleftrightarrow\, s_{H'}(\gamma) \in \omega(\Lambda') +W'\Longleftrightarrow\, \gamma\in \Lambda_1. \end{align*} $$

Therefore, for every $\Lambda '\in \Omega ^0_{\Lambda }\cap \Xi ^\Gamma $ such that $\pi _1(\Lambda ')=\Lambda _1$ we have

(5.8) $$ \begin{align} \Lambda'\subseteq \Lambda_1. \end{align} $$

On the other hand, fix $\gamma $ in $\Lambda _1$ . By (5.6), for every $\Lambda '\kern1.3pt{\in}\kern1.3pt \Omega ^0_{\Lambda }\kern1.3pt{\cap}\kern1.3pt \Xi ^\Gamma $ such that $\pi _1(\Lambda ')\kern1.3pt{=}\kern1.3pt\Lambda _1$ we have

$$ \begin{align*}s_{H'}(\gamma)\in \omega(\Lambda') + W' \Leftrightarrow \omega(\Lambda') \in \omega(\Xi_{\Lambda}+\gamma).\end{align*} $$

Thus, there is $\Lambda "$ in $\Xi _{\Lambda }+\gamma \subseteq \Xi ^\Gamma $ such that $\omega (\Lambda ")=\omega (\Lambda ')$ . Then $\Lambda "-\gamma $ is in $\Xi _{\Lambda }$ , and thus $\gamma $ is in $\Lambda "$ . Therefore,

(5.9) $$ \begin{align} \Lambda_1 \subseteq \bigcup_{\Lambda"\in \Omega^0_\Lambda \cap \Xi^\Gamma\text{ s.t. } \omega(\Lambda") = \omega(\Lambda')} \Lambda". \end{align} $$

Observe that for every $\Lambda '\in \Omega ^0_{\Lambda }\cap \Xi ^\Gamma $ and every $\Lambda " \in \Xi ^\Gamma $ such that $\omega (\Lambda ')=\omega (\Lambda ")$ we have that $\widetilde {\pi }(\Lambda ')=\widetilde {\pi }(\Lambda ")$ , and thus $\Lambda "\in \Omega ^0_{\Lambda }\cap \Xi ^\Gamma $ . In particular, $\pi _1(\Lambda ')=\pi _1(\Lambda ")$ , which, together with (5.9), implies

(5.10) $$ \begin{align} \Lambda_1 \subseteq \bigcup_{\pi_1(\Lambda") = \Lambda_1} \Lambda". \end{align} $$

We now prove that for every $\Lambda '\in \Omega ^0_{\Lambda }\cap \Xi ^\Gamma $ and every $\Lambda " \in \Omega ^0_\Lambda $ such that $\pi _1(\Lambda ')=\pi _1(\Lambda ")$ , we have that

(5.11) $$ \begin{align} \Lambda"\in \Xi^\Gamma. \end{align} $$

First, observe that for all $\Lambda '$ and $\Lambda "$ in $\Omega ^0_\Lambda $ we have that $\pi _1(\Lambda ')=\pi _1(\Lambda ") \Leftrightarrow \widetilde {\pi }(\Lambda ") = \widetilde {\pi }(\Lambda ')$ . Now let $\Lambda '\in \Omega ^0_{\Lambda }\cap \Xi ^\Gamma $ and $\Lambda " \in \Omega ^0_\Lambda $ be such that $\widetilde {\pi }(\Lambda ") = \widetilde {\pi }(\Lambda ')$ . By definition of $\widetilde {\pi }$ , this holds if and only if there exists t in $\mathbb {R}^d$ such that $\Lambda "-t\in \Xi ^\Gamma $ and $[(-t,\omega (\Lambda "-t))]=[(0,\omega (\Lambda '))]$ , which is equivalent to the existence of $\gamma $ in $\Gamma $ such that

$$ \begin{align*}(-t,\omega(\Lambda"-t)) - (0,\omega(\Lambda')) = (\gamma,s_{H'}(\gamma)).\end{align*} $$

Then $-t = \gamma \in \Gamma $ and we get $\Lambda "\subseteq \Gamma -\gamma =\Gamma $ , which proves (5.11). By (5.8), (5.10) and (5.11), we conclude that

$$ \begin{align*} \Lambda_1 = \bigcup_{\pi_1(\Lambda")=\Lambda_1} \Lambda", \end{align*} $$

which is equivalent to

(5.12) $$ \begin{align} \Lambda_1 = \bigcup_{\Lambda'\in\; \widetilde{\pi}^{-1}(\pi_0(\Lambda_1))} \Lambda'. \end{align} $$

If $\Lambda _1$ is not in $\pi _1(\Omega _\Lambda ^0\cap \Xi ^\Gamma )$ then there is t in $\mathbb {R}^d$ such that $\Lambda _1-t$ is in $\pi _1(\Omega _\Lambda ^0\cap \Xi ^\Gamma )$ . By (5.12), we have that

$$ \begin{align*} \Lambda_1-t = \bigcup_{\widetilde{\Lambda}\in\; \widetilde{\pi}^{-1}(\pi_0(\Lambda_1-t))} \widetilde{\Lambda}.\end{align*} $$

Since $\widetilde {\pi }(\widetilde {\Lambda })= \pi _0(\Lambda _1-t)$ if and only if $\widetilde {\pi }(\widetilde {\Lambda }-(-t))= \pi _0(\Lambda _1)$ , we conclude that

(5.13) $$ \begin{align} \Lambda_1 = \bigcup_{\widetilde{\Lambda}\in\; \widetilde{\pi}^{-1}(\pi_0(\Lambda_1-t))} \widetilde{\Lambda} - (-t) = \bigcup_{\Lambda'\in\; \widetilde{\pi}^{-1}(\pi_0(\Lambda_1))} \Lambda', \end{align} $$

which finishes the proof of (5.1).

5.2.4 $(\Omega _\Lambda ,\mathbb {R}^d)$ almost automorphic extension of $(\Omega _{\mathrm {MS}, \mathrm {me}},\mathbb {R}^d)$

Finally, suppose that $\widetilde {\pi }$ is an almost automorphic extension of $(\Omega _{\mathrm {MS}, \mathrm {me}},\mathbb {R}^d)$ . By [Reference VeechVee70, Lemma 4.1], we have that the set

$$ \begin{align*}\widetilde{\pi}(R_{\widetilde{\pi}}(\Omega_\Lambda)) = \{x\in\Omega_{\mathrm{MS}, \mathrm{me}}\mid\widetilde{\pi}^{-1}(x)\;\textrm{is a singleton}\}\end{align*} $$

is a residual set in $\Omega _{\mathrm {MS}, \mathrm {me}}$ , and by [Reference AujogueAuj16a], the set $\pi _0(R_{\pi _0}(\Omega _{\mathrm {MS}}))$ is also a residual set in $\Omega _{\mathrm {MS}, \mathrm {me}}$ . Then

$$ \begin{align*} \pi_0(R_{\pi_0}(\Omega_{\mathrm{MS}}))\cap\widetilde{\pi}(R_{\widetilde{\pi}}(\Omega_\Lambda)) \end{align*} $$

is also a residual set in $\Omega _{\mathrm {MS}, \mathrm {me}}$ . By (5.13), for every $\Lambda _1$ in $R_{\pi _0}(\Omega _{\mathrm {MS}})$ such that $\pi _0(\Lambda _1)\in \widetilde {\pi }(R_{\widetilde {\pi }}(\Omega _\Lambda ))$ we have that $\Lambda _1$ is in $\Omega ^0_\Lambda $ , which concludes the proof of the main technical lemma $^{\prime }$ .