boxdefDefinition[section]

# Inter model sets in ${\mathbb{R}}^{d}$ are model sets

Christoph Richard  Department für Mathematik, Friedrich-Alexander-Universität
Erlangen-Nürnberg, Cauerstrasse 11, 91058 Erlangen, Germany
<EMAIL_ADDRESS>and  Nicolae Strungaru  Department of Mathematical
Sciences, MacEwan University
10700–104 Avenue, Edmonton, AB, T5J 4S2
and
Institute of Mathematics “Simon Stoilow”
Bucharest, Romania<EMAIL_ADDRESS>
###### Abstract.

We show that any translate of a model set is a model set in some modified cut-
and-project scheme. Restricting to Euclidean direct space, we show that any
translate of an inter model set is a model set in some modified cut-and-
project scheme with second countable internal space. In both cases, the window
in the modified cut-and-project scheme inherits the topological and measure-
theoretic properties of the original window. Our results hold in fact for a
class beyond inter model sets, which we call almost model sets.

##  1\. Motivation

A _cut-and-project scheme_ $(G,H,{\mathcal{L}})$ consists of locally compact
abelian groups $G,H$ and a discrete and cocompact subgroup
${\mathcal{L}}\subseteq G\times H$, also called _lattice_ , such that
${\mathcal{L}}$ projects injectively to $G$ and densely to $H$. Sometimes $G$
is called _direct space_ , and $H$ is called _internal space_. Denote the
coordinate projections by $\pi^{G}$ and $\pi^{H}$, respectively. Given any
_window_ $W\subseteq H$, the _projection set_ $\Lambda_{W}\subseteq G$ is
defined by $\Lambda_{W}=\pi^{G}({\mathcal{L}}\cap(G\times W))$. Often the
abbreviation $L=\pi^{G}({\mathcal{L}})$ is used. Since
$\pi^{G}|_{{\mathcal{L}}}:{\mathcal{L}}\to G$ is one-to-one, the so-called
_star-map_ $\star:L\to H$, given by
$\star=\pi^{H}\circ(\pi^{G}|_{{\mathcal{L}}})^{-1}$, is well-defined. Using
the $\star$-map, we may conveniently write
${\mathcal{L}}=\\{(g,g^{\star}):g\in L\\}$ and $\Lambda_{W}=\\{g\in
L:g^{\star}\in W\\}$. If $W$ is relatively compact and (Borel) measurable, we
call $\Lambda_{W}$ a _weak model set_. If additionally $W$ admits interior
points, then $\Lambda_{W}$ is called a _model set_. If additionally $W$ has a
topological boundary $\partial W=\overline{W}\setminus W^{\circ}$ of zero Haar
measure, then $W$ is called measure-theoretically regular, and $\Lambda_{W}$
is a called a _regular model set_. Background about model sets can be found
e.g. in [2, 18].

A class of examples in $G={\mathbb{R}}$ arises from unimodular Pisot
substitutions in one dimension with fixed point $\Lambda\subset{\mathbb{R}}$,
see [4, Sec. 3] for a recent exposition. In that setting, the substitution is
used in order to construct a cut-and-project scheme
$({\mathbb{R}},H,{\mathcal{L}})$ and an iterated function system in $H$ such
that $\Lambda\subseteq\Lambda_{W}$, where the compact set $W\subseteq H$ is
the attractor of the iterated function system. Under additional assumptions on
the substitution we may have
$\Lambda_{W^{\circ}}\subseteq\Lambda\subseteq\Lambda_{W}$, where $W$ is
measure-theoretically regular. In that case we call $\Lambda$ an _inter
regular model set_ , compare [13, 15]. Then $\Lambda$ and the regular model
set $\Lambda_{W}$ coincide up to mismatches of zero density, and they have the
same spectral properties. In abuse of notation, some authors call any
translate of such $\Lambda$ a regular model set [15]. We will study the
question whether an inter regular model set may be a regular model set, in
some possibly modified cut-and-project scheme.

For illustration consider the Fibonacci substitution $a\mapsto ab$, $b\mapsto
a$, compare [2, Ex. 4.6]. Starting with the seed $a|a$, the squared Fibonacci
distribution has a unique fixed point. Now replace $a$ and $b$ by intervals of
lengths $\tau$ and $1$, respectively, where $\tau=(1+\sqrt{5})/2$ is the
golden mean, center the seed at the origin, and consider the collection
$\Lambda\subset{\mathbb{Z}}[\tau]$ of all left interval endpoints. A cut-and-
project scheme [2, Ex. 7.3] is obtained from the Minkowski embedding of
${\mathbb{Z}}[\tau]$, for which $\Lambda$ is a regular model set with half
closed interval as window, while the attractor of the iterated function system
gives the closure of the window. As the associated cut-and-project scheme has
an injective $\star$-map, any inter regular model set in the above cut-and-
project scheme is in fact a regular model set in the same cut-and-project
scheme. A corresponding measure-theoretically regular window is easily
constructed, see the proof of Proposition 2.4.

Consider now a weak model set $\Lambda_{W}$ in a general cut-and-project
scheme $(G,H,{\mathcal{L}})$. One may pose the question whether there exists a
modified cut-and-project scheme $(G,H^{\prime},{\mathcal{L}}^{\prime})$ with
injective $\star$-map such that
$\Lambda_{W}=\pi^{G}({\mathcal{L}}^{\prime}\cap(G\times W^{\prime}))$ for some
window $W^{\prime}\subseteq H^{\prime}$, where $W^{\prime}$ inherits the
following properties $(T)$ and $(M)$ from the window $W$:

  * •

_topological properties (T)_

    * –

precompactness: $\overline{W}$ is compact in $H$

    * –

existence of interior points: $W^{\circ}\neq\varnothing$

    * –

topological regularity: $\overline{W}=\overline{W^{\circ}}$

  * •

_measure-theoretic properties (M)_

    * –

measure-theoretic regularity: $\partial W=\overline{W}\setminus W^{\circ}$ has
zero Haar measure

A partial answer has already been given in [14, Lem. 4], where the
construction of a modified cut-and-project scheme uses the Bohr
compactification of $G$. While the construction preserves property $(T)$,
property $(M)$ may be lost. Also, the construction does not preserve
metrisability of internal space. Restricting to Euclidean direct space and
second countable internal space, we give an affirmative answer to the above
question in Proposition 2.6.

Inter model sets also arise naturally from a dynamical analysis of model sets.
To explain this, identify a point set $\Lambda\subseteq G$ by its Dirac comb
$\sum_{g\in\Lambda}\delta_{g}$ and consider its _hull_ $\mathbb{X}(\Lambda)$,
i.e., the translation orbit closure of $\Lambda$ with respect to the vague
topology. Assume for the following that $\Lambda=\Lambda_{W}$ is a model set.
Then any $\Gamma\in\mathbb{X}(\Lambda)$ satisfies
$\Lambda_{t+W^{\circ}}\subseteq s+\Gamma\subseteq\Lambda_{t+\overline{W}}$ for
some $(s,t)\in G\times H$, compare Lemma 4.1 below. Whereas such $\Gamma$ is
called an inter model set [6, Eq. (7)], we will prefer the name _shifted inter
model set_ , and we will speak of an _inter model set_ if $s=0$. One may ask
whether any shifted inter model set is an inter model set, possibly in some
different cut-and-project scheme. We will give an affirmative answer in
Theorem 2.2 below.

Note that $(s,t)$ is uniquely determined modulo ${\mathcal{L}}$ if the window
is non-empty and aperiodic, i.e., $t+W=W$ implies $t=0$, compare [6]. In that
case any $\Gamma\in\mathbb{X}(\Lambda)$ can be parametrised by
$(s,t)+{\mathcal{L}}$. This gives rise to the so-called torus parametrisation,
see [9, 10] for a recent approach. Again, the question arises whether an
element of $\mathbb{X}(\Lambda)$ may be a (shifted) model set. If the
translated window $t+W$ is in generic position, i.e., if the boundary of $t+W$
does not intersect the projected lattice, then we have
$\Lambda_{t+W^{\circ}}=\Lambda_{t+\overline{W}}=\Lambda_{t+W}$. Therefore, if
$\Gamma$ is parametrised by $(s,t)+{\mathcal{L}}$ for $t+W$ in generic
position, $s+\Gamma=\Lambda_{t+W}$ implies that $\Gamma$ is a shifted model
set and hence a model set. We show in Proposition 4.2 that the latter
conclusion holds for all $\Gamma\in\mathbb{X}(\Lambda)$.

##  2\. Setting and main results

In order to formulate our results, we use the following setting. We allow for
a shift of the lattice in the cut-and-project construction, which leads to
_shifted projection sets_ defined by

$\Lambda_{W}(x)=\pi^{G}((x+{\mathcal{L}})\cap(G\times W))\ .$

Note that shifting the lattice is equivalent to translating both the
projection set and the window. Indeed, for $x=(s,t)\in G\times H$ we have
$\Lambda_{W}(x)=s+\Lambda_{-t+W}$. Recall that, in abuse of notation,
$\Lambda_{W}(x)$ is sometimes called a model set if $\Lambda_{W}$ is a model
set. In a given cut-and-project scheme $(G,H,{\mathcal{L}})$ with window
$W\subseteq H$, one may consider a _shifted inter projection set_
$\Gamma\subseteq G$ satisfying

$\Lambda_{W^{\circ}}(x)\subseteq\Gamma\subseteq\Lambda_{\overline{W}}(x)$

for some $x\in G\times H$.

######  Remark 2.1.

As discussed in the previous section, shifted inter projection sets arise
naturally when studying projection sets from a dynamical perspective. If
$\Lambda_{W}$ is a projection set, then every
$\Gamma\in\mathbb{X}(\Lambda_{W})$ is a shifted inter projection set. A proof,
which extends previous results from [21, 6, 9], will be given in Lemma 4.1
below.

We argue first that shifting is irrelevant, i.e., any translate of an inter
model set is an inter model set in some modified cut-and-project scheme, such
that properties $(T)$ and $(M)$ of the window are preserved. This is the
content of the following statement.

######  Theorem 2.2.

Consider a cut-and-project scheme $(G,H,{\mathcal{L}})$ and some $a\in G$.
Then there exist a cut-and-project scheme
$(G,H^{\prime},{\mathcal{L}}^{\prime})$ and some $b\in H^{\prime}$ such that

  * (a)

$H$ is an open and closed subgroup of $H^{\prime}$.

  * (b)

$(a,b)\in{\mathcal{L}}^{\prime}$.

  * (c)

${\mathcal{L}}^{\prime}\cap(G\times H)={\mathcal{L}}$.

  * (d)

For each $W\subseteq H$ we have
$\Lambda_{W}=\pi^{G}({\mathcal{L}}^{\prime}\cap(G\times W))$ and
$a+\Lambda_{W}=\pi^{G}({\mathcal{L}}^{\prime}\cap(G\times(b+W)))$.

  * (e)

Both $W\subseteq H^{\prime}$ and $b+W\subseteq H^{\prime}$ inherit any
property in $(T)$ or $(M)$ from $W\subseteq H$.

  * (f)

$H^{\prime}$ inherits metrisability from $H$.

######  Remark 2.3.

As a consequence of the above theorem, any shifted weak model set, shifted
model set, shifted regular model set and shifted inter model set,
respectively, is a weak model set, model set, regular model set and inter
model set, respectively. Moreover, the same will be true for almost regular
model sets which we introduce after Eqn. (2.2): shifted almost regular model
sets are almost regular model sets.

Note that the projection set in the above proposition may arise from a
different cut-and-project scheme as the shifted projection set. We will study
so-called translation cut-and-project schemes in Section 3, where the above
result is a direct consequence of Proposition 3.2. Our construction is
inspired by [11, Thm. 6.1].

One may further ask whether an inter projection set may be a projection set,
such that properties $(T)$ and $(M)$ of the window are preserved. As pointed
out above, the answer is affirmative for cut-and-project schemes where
${\mathcal{L}}$ projects injectively to $H$, meaning that the $\star$-map of
the cut-and-project scheme is one-to-one. Indeed, we have:

######  Proposition 2.4.

Let $(G,H,{\mathcal{L}})$ be a cut-and-project scheme with injective
$\star$-map. Then any inter projection set in $(G,H,{\mathcal{L}})$ is a
projection set in $(G,H,{\mathcal{L}})$. Moreover the window of the projection
set inherits properties $(T)$ and $(M)$. Assume additionally that $G$ is
$\sigma$-compact and that the window $W$ of the inter projection set is
precompact and measurable. Then the window of the projection set is precompact
and measurable.

###### Proof.

Consider a cut-and-project scheme $(G,H,{\mathcal{L}})$ such that
${\mathcal{L}}$ projects injectively to $H$. Consider $\Gamma\subseteq G$ and
$W\subseteq H$ such that
$\Lambda_{W^{\circ}}\subseteq\Gamma\subseteq\Lambda_{\overline{W}}$. Due to
injectivity of the $\star$-map, we have $\Gamma=\Lambda_{\Gamma^{\star}}$ for
any $\Gamma\subseteq L$. Then $W^{\prime}=W^{\circ}\cup\Gamma^{\star}$
satisfies
$\Lambda_{W^{\prime}}=\Lambda_{W^{\circ}}\cup\Lambda_{\Gamma^{\star}}=\Lambda_{W^{\circ}}\cup\Gamma=\Gamma$.
Let us analyse the window properties $(T)$ and $(M)$. Note that the definition
of $W^{\prime}$ and
$\Lambda_{W^{\circ}}\subseteq\Gamma\subseteq\Lambda_{\overline{W}}$ give
$W^{\circ}\subseteq W^{\prime}\subseteq\overline{W}$, which implies

(2.1) $W^{\circ}\subseteq(W^{\prime})^{\circ}\subseteq
W^{\prime}\subseteq\overline{W^{\prime}}\subseteq\overline{W}\ .$

Assume that $W$ is precompact. Then $\overline{W}$ is compact. Therefore
$W^{\prime}$ is precompact by (2.1).

Assume $W^{\circ}\neq\varnothing$. Then $(W^{\prime})^{\circ}\neq\varnothing$
follows from (2.1).

Assume $\overline{W}=\overline{W^{\circ}}$. Together with (2.1) this gives
$\overline{W}=\overline{W^{\circ}}\subseteq\overline{W^{\prime}}\subseteq\overline{W}$.
We thus can conclude
$\overline{W}=\overline{W^{\prime}}=\overline{W^{\circ}}\subseteq\overline{(W^{\prime})^{\circ}}$,
and the claim follows.

Assume that $W$ is measure-theoretically regular. By (2.1) we have
$\partial(W^{\prime})\subseteq\overline{W}\backslash W^{\circ}=\partial W$.
Thus $W^{\prime}$ is measure-theoretically regular.

Finally, assume that $G$ is $\sigma$-compact and that $W$ is precompact and
measurable. Then $W^{\prime}$ is precompact by the previous argument, and
$W^{\prime}$ measurable as $\Gamma$ is countable, due to uniform discreteness
of $\Lambda_{\overline{W}}$ and $\sigma$-compactness of $G$. ∎

######  Remark 2.5.

Examples of setting in Proposition 2.4 are complete cut-and-project schemes,
i.e., cut-and-project schemes where ${\mathcal{L}}$ projects both injectively
to $H$ and densely to $H$. For $G=\mathbb{{\mathbb{R}}}^{d}$ and
$H=\mathbb{{\mathbb{R}}}^{m}\times\mathbb{T}^{n}\times\mathbb{D}$, where
$\mathbb{D}$ is a countable discrete group, complete cut-and-project schemes
have recently been classified [1, Thm. 3].

In Section 6 we study when a projection set may arise from a cut-and-project
scheme $(G,H,{\mathcal{L}})$ with injective $\star$-map. We will construct
such a cut-and-project scheme that also preserves properties $(T)$ and $(M)$
of a window. Our construction assumes Euclidean direct space $G$ and assumes
$H$ to be second countable. The following result is a reformulation of Theorem
6.5 below.

######  Proposition 2.6.

Consider a cut-and-project scheme $({\mathbb{R}}^{d},H,{\mathcal{L}})$ with
second countable internal space $H$. Then any inter projection set in
$({\mathbb{R}}^{d},H,{\mathcal{L}})$ is a projection set, and the window of
the projection set inherits properties $(T)$ and $(M)$ from the window of the
inter projection set. Assume additionally that the window $W$ related to the
inter projection set is precompact and measurable. Then the window of the
projection set is precompact and measurable.

In order to refine the above result, we investigate when the internal space
can be assumed to be second countable. This is indeed the case for regular
model sets. In fact, we prove a stronger statement in Proposition 5.1 below,
by enlarging the class of inter model sets as follows.

######  Definition 2.7.

We say that $\Gamma\subseteq G$ is an _almost model set_ if there exist a cut-
and-project scheme $(G,H,{\mathcal{L}})$ and sets $U\subseteq W\subseteq H$
such that $U$ is non-empty open, $W$ is compact, and

(2.2) $\Lambda_{U}\subseteq\Gamma\subseteq\Lambda_{W}\ .$

If additionally $W\backslash U$ has vanishing Haar measure, we call $\Gamma$
an _almost regular model set_. We will also refer to $\Gamma$ as being a
_$(U,W)$ -almost model set_.

######  Remark 2.8.

  * (a)

If $\Gamma$ is an inter (regular) model set, then $\Gamma$ is an almost
(regular) model set.

  * (b)

For an $(U,W)$-almost model set $\Gamma$, the density formula for weak model
sets [8, Prop. 3.4] combined with (2.2) gives

$\begin{split}\operatorname{dens}({\mathcal{L}})&\cdot
m_{H}(U)\leq\underline{\operatorname{dens}}(\Lambda_{U})\leq\underline{\operatorname{dens}}(\Gamma)\\\
&\leq\overline{\operatorname{dens}}(\Gamma)\leq\overline{\operatorname{dens}}(\Lambda_{W})\leq\operatorname{dens}({\mathcal{L}})\cdot
m_{H}(W)\ .\end{split}$

Here $m_{H}$ denotes a Haar measure on $H$,
$\operatorname{dens}({\mathcal{L}})$ denotes the inverse measure of a
measurable fundamental domain of ${\mathcal{L}}$, and
$\underline{\operatorname{dens}}$, $\overline{\operatorname{dens}}$ denote the
lower and upper density along a given averaging sequence. In particular, any
almost regular model set $\Gamma$ differs from the regular model set
$\Lambda_{W}$ on a set of density zero. This readily implies that any almost
regular model set $\Gamma$ is Weyl almost periodic [17], has the same
autocorrelation, diffraction and Fourier–Bohr coefficients as $\Lambda_{W}$,
and that its hull, equipped with its natural translation action and its unique
ergodic probability measure, is measure-theoretically isomorphic to that of
$\Lambda_{W}$.

We can now state our main result, whose proof will be given at the end of
Section 6.

######  Theorem 2.9.

Let $\Lambda\subseteq{\mathbb{R}}^{d}$ be any shifted almost model set from
some cut-and-project scheme. Then, $\Lambda$ is a model set in some modified
cut-and-project scheme with second countable internal space and injective
$\star$-map, such that measurability and properties $(T)$ and $(M)$ of the
window are preserved. In particular, any translate of an inter (regular) model
set in Euclidean space is a (regular) model set.

Let us point out an application of Theorem 2.9 which is related to point sets
obtained from substitution. We cite Theorem 5.3 from [13], where we restrict
to Delone sets instead of Delone multisets for convenience. Recall that a
Delone set is a point set that is uniformly discrete and relatively dense. For
the notion of finite local complexity, see the remark before Proposition 4.5.

######  Theorem 2.10 ([13]).

Let $\Lambda\subset{\mathbb{R}}^{d}$ be a primitive substitution Delone set
such that every $\Lambda$-cluster is legal and such that $\Lambda$ has finite
local complexity. Then $\Lambda$ has pure point dynamical spectrum if and only
if $\Lambda$ is a shifted inter model set with topologically regular window. ∎

Due to our results, we may replace “shifted inter model set” by “model set”.

##  3\. Translation cut-and-project schemes

Consider a projection set $\Lambda_{W}$ in some cut-and-project scheme
$(G,H,{\mathcal{L}})$ with window $W\subseteq H$. Given some $a\in G$, we ask
whether $a+\Lambda_{W}$ is a projection set. The answer is affirmative for
$a\in\pi_{G}({\mathcal{L}})$, as we then have
$a+\Lambda_{W}=\Lambda_{a^{\star}+W}$. In general $a+\Lambda_{W}$ is a
projection set in some modified cut-and-project set
$(G,H^{\prime},{\mathcal{L}}^{\prime})$, where $H^{\prime}$ is a factor group
of $H\times\mathbb{Z}$. Our construction is inspired by [11, Thm. 6.1]. Fix
nonzero $a\in G$ and consider the group $\langle
a\rangle=\mathrm{Span}_{{\mathbb{Z}}}(\\{a\\})$ generated by $a$. We first
consider the case $\langle a\rangle\cap\pi_{G}({\mathcal{L}})=\\{0\\}$, which
we call the incommensurate case.

###  3.1. The incommensurate case

A natural construction leads to the following result.

######  Proposition 3.1 (translation CPS).

Consider a cut-and-project scheme $(G,H,{\mathcal{L}})$. Fix nonzero $a\in G$
so that $\langle a\rangle\cap\pi_{G}({\mathcal{L}})=\\{0\\}$. Define
$H^{\prime}=H\times{\mathbb{Z}}$ and
${\mathcal{L}}^{\prime}=({\mathcal{L}}\times\\{0\\})+\langle(a,0,1)\rangle$.
Then the following hold.

  * (a)

$(G,H^{\prime},{\mathcal{L}}^{\prime})$ is a cut-and-project scheme. If
${\mathcal{L}}$ is countable, then so is ${\mathcal{L}}^{\prime}$.

  * (b)

For $W\subseteq H$ and $n\in{\mathbb{Z}}$ we have
$na+\Lambda_{W}=\Lambda^{\prime}_{W^{\prime}}$, where
$W^{\prime}=W\times\\{n\\}$.

  * (c)

$H$ is homeomorphic to $H\times\\{0\\}$ and $H\times\\{n\\}$ is open and
closed in $H^{\prime}$.

  * (d)

The window $W^{\prime}=(W\times\\{n\\})\subseteq H^{\prime}$ inherits
properties $(T)$ and $(M)$ from $W\subseteq H$. If $W$ is measurable, then
$W^{\prime}$ is measurable.

###### Proof.

“(a)” By definition, countability of ${\mathcal{L}}^{\prime}$ follows from
countability of ${\mathcal{L}}$. It is clear that ${\mathcal{L}}^{\prime}$ is
a group. In order to show that ${\mathcal{L}}^{\prime}$ is a lattice, it
suffices to show that ${\mathcal{L}}^{\prime}$ is a Delone set. Since
${\mathcal{L}}$ is a lattice in $G\times H$, there exists a compact set
$K\subseteq G\times H$ such that ${\mathcal{L}}+K=G\times H$. This implies

${\mathcal{L}}^{\prime}+(K\times\\{0\\})=({\mathcal{L}}\times\\{0\\})+\langle(a,0,1)\rangle+(K\times\\{0\\})=(G\times
H\times\\{0\\})+\langle(a,0,1)\rangle=G\times H\times\mathbb{Z}=G\times
H^{\prime}\ .$

Since ${\mathcal{L}}$ is uniformly discrete, there exists a zero neighborhood
$U\subseteq G\times H$ such that ${\mathcal{L}}\cap U=\\{(0,0)\\}$. Then the
open zero neighborhood $U\times\\{0\\}\subseteq G\times H^{\prime}$ satisfies

${\mathcal{L}}^{\prime}\cap(U\times\\{0\\})=(({\mathcal{L}}\times\\{0\\})+\langle(a,0,1)\rangle)\cap(U\times\\{0\\})=({\mathcal{L}}\times\\{0\\})\cap(U\times\\{0\\})=\\{(0,0,0)\\}\
.$

We show next that $\pi^{G}:G\times H^{\prime}\to G$ is one-to-one when
restricted to ${\mathcal{L}}^{\prime}$. Assume
$\ell^{\prime}=(\ell,0)+n(a,0,1)\in{\mathcal{L}}^{\prime}$ for some
$\ell\in{\mathcal{L}}$ and $n\in{\mathbb{Z}}$ such that
$\pi^{G}(\ell^{\prime})=na+\pi^{G}(\ell)=0$. By assumption, we get $n=0$ and
$\pi^{G}(\ell)=0$. As $\pi^{G}:G\times H\to G$ is one-to-one when restricted
to ${\mathcal{L}}$, we get $\ell=0$, which implies
$\ell^{\prime}=(\ell,0)+n(a,0,1)=(0,0,0)$. This proves the claim.

Denseness of $\pi^{H^{\prime}}({\mathcal{L}}^{\prime})$ in $H^{\prime}$ is
inherited from $(G,H,{\mathcal{L}})$. Indeed, consider any non-empty open set
$U\subseteq H^{\prime}$. Then there exist some non-empty open set $V\subseteq
H$ and $n\in{\mathbb{Z}}$ such that $V\times\\{n\\}\subseteq U$. Since
$(G,H,{\mathcal{L}})$ is a cut-and-project scheme, there exists some
$\ell\in{\mathcal{L}}$ such that $\pi^{H}(\ell)\in V$. We then have
$\pi^{H^{\prime}}((\ell,0)+n(a,0,1))=(\pi^{H}(\ell),0)+(0,n)=(\pi^{H}(\ell),n)\in\pi^{H^{\prime}}({\mathcal{L}}^{\prime})\cap(V\times\\{n\\})\subseteq\pi^{H^{\prime}}({\mathcal{L}}^{\prime})\cap
U$. This proves (a).

“(b)” follows by direct calculation. Let $W^{\prime}=W\times\\{n\\}\in
H^{\prime}$. Then

$\begin{split}\Lambda^{\prime}_{W^{\prime}}&=\pi^{G}({\mathcal{L}}^{\prime}\cap(G\times
W^{\prime}))=\pi^{G}((({\mathcal{L}}\times\\{0\\})+\langle(a,0,1)\rangle)\cap(G\times
W\times\\{n\\}))\\\
&=\pi^{G}((({\mathcal{L}}\times\\{0\\})+n(a,0,1))\cap(G\times
W\times\\{n\\}))=\pi^{G}({\mathcal{L}}\cap(G\times W))+na=\Lambda_{W}+na\
.\end{split}$

“(c)” is obvious, and “(d)” follows from “(c)”. ∎

###  3.2. The general case

The lattice ${\mathcal{L}}^{\prime}$ in the above construction does no longer
project injectively to $G$ if $\langle
a\rangle\cap\pi_{G}({\mathcal{L}})\neq\\{0\\}$. Indeed, let $m\in\mathbb{N}$
be minimal such that $b=ma\in\pi_{G}({\mathcal{L}})$, and consider arbitrary
$(na+x,x^{\star},n)\in{\mathcal{L}}^{\prime}$. Then $na+x=0$ implies $x=sb$
and $n=-sm$ for some $s\in{\mathbb{Z}}$. We thus obtain
$(na+x,x^{\star},n)=(0,sb^{\star},-sm)$, which violates injectivity. But
injectivity might be restored from factoring by the subgroup
$J=\langle(b^{\star},-m)\rangle\subseteq H^{\prime}$. Consider thus the factor
group $H^{\prime}/J$ equipped with the quotient topology. Take any $(h,n)\in
H^{\prime}$. Writing $n=r+sm$ for $r\in\\{0,\ldots,m-1\\}$ and
$s\in\mathbb{Z}$ and noting $(-sb^{\star},sm)\in J$, we infer
$(h,n)+J=(h+sb^{\star},r)+J$. This means that the factor group $H^{\prime}/J$
is group-theoretically isomorphic to $(H\times{\mathbb{Z}}_{m},\oplus)$, where

$(h_{1},r_{1})\oplus(h_{2},r_{2})=\left\\{\begin{array}[]{cc}(h_{1}+h_{2},r_{1}+r_{2})&\mbox{
if }r_{1}+r_{2}<m\\\ (h_{1}+h_{2}+b^{\star},r_{1}+r_{2}-m)&\mbox{ if
}r_{1}+r_{2}\geq m\\\ \end{array}\right.\ .$

Moreover, under the above identification, the induced topology on
$H\times{\mathbb{Z}}_{m}$ is the product topology.

The previous discussion suggests that we might obtain a translation cut-and-
project scheme for the general case by a factor group construction. Quotient
cut-and-project schemes have been intensively studied in [10, Sec. 6] by
factoring with compact subgroups. Here the situation is different as
$J\subseteq H^{\prime}$ may not be compact. Whereas most arguments from [10]
apply in the present case, uniform discreteness of the quotient lattice has to
be shown separately. For the ease of the reader, we provide complete
arguments.

######  Proposition 3.2 (extended quotient CPS).

Let $(G,H,{\mathcal{L}})$ be a cut-and-project scheme. Consider nonzero $a\in
G$. If $\langle a\rangle\cap\pi_{G}({\mathcal{L}})\neq\\{0\\}$, let
$m\in\mathbb{N}$ be minimal so that $b=ma\in\pi_{G}({\mathcal{L}})$. Otherwise
set $m=0$ and $b=0$. Consider the group
$J=\langle(b^{\star},-m)\rangle\subseteq H\times{\mathbb{Z}}$ equipped with
the discrete topology. Define $H^{\prime}=(H\times{\mathbb{Z}})/J$ and
${\mathcal{L}}^{\prime}=({\mathcal{L}}\times\\{0\\})+\langle(a,0,1)\rangle+(\\{0\\}\times
J)\subseteq G\times H^{\prime}$. Then the following hold.

  * (a)

$(G,H^{\prime},{\mathcal{L}}^{\prime})$ is a cut-and-project scheme. If
${\mathcal{L}}$ is countable, then so is ${\mathcal{L}}^{\prime}$.

  * (b)

For $W\subseteq H$ and $n\in{\mathbb{Z}}$ we have
$na+\Lambda_{W}=\Lambda^{\prime}_{W^{\prime}}$, where
$W^{\prime}=(W\times\\{n\\})+J$.

  * (c)

The map $x\mapsto(x,0)+J$ is a continuous open embedding of $H$ into
$H^{\prime}$.

  * (d)

The window $W^{\prime}=((W\times\\{n\\})+J)\subseteq H^{\prime}$ inherits
properties $(T)$ and $(M)$ from $W\subseteq H$. If $W$ is measurable, then
$W^{\prime}$ is measurable.

######  Remark 3.3.

Note that Proposition 3.1 is a special case of Proposition 3.2 with trivial
group $J$. This justifies the same notation for the two cut-and-project
schemes.

###### Proof.

We use the canonical identification $(G\times
H\times{\mathbb{Z}})/(\\{0\\}\times J)\cong G\times H^{\prime}$, compare [10,
Sec. 6].

“(a)” By definition, countability of ${\mathcal{L}}^{\prime}$ follows from
countability of ${\mathcal{L}}$. We start by showing that
${\mathcal{L}}^{\prime}$ is a lattice in $G\times H^{\prime}$. Note first that
${\mathcal{L}}^{\prime}$ is a group as
${\mathcal{L}}^{\prime}={\mathcal{L}}+\langle(a,0,1)\rangle+(\\{0\\}\times J)$
and as ${\mathcal{L}}$ is a group. The group ${\mathcal{L}}^{\prime}$ is also
relatively dense: Take $K\in G\times H$ such that ${\mathcal{L}}+K=G\times H$,
and define $K^{\prime}=(K\times\\{0\\})+(\\{0\\}\times J)$. We then have

$\begin{split}{\mathcal{L}}^{\prime}+K^{\prime}&=({\mathcal{L}}+\langle(a,0,1)\rangle+(\\{0\\}\times
J))+((K\times\\{0\\})+(\\{0\\}\times J))\\\
&={\mathcal{L}}+\langle(a,0,1)\rangle+(K\times\\{0\\})+(\\{0\\}\times J)\\\
&=(G\times H\times\\{0\\})+\langle(a,0,1)\rangle+(\\{0\\}\times J)=G\times
H^{\prime}\ .\end{split}$

Next, we prove uniform discreteness of ${\mathcal{L}}^{\prime}$. Since
${\mathcal{L}}$ is a lattice in $G\times H$ there exists a zero neighborhood
$U\subseteq G\times H$ such that ${\mathcal{L}}\cap U=\\{(0,0)\\}$. Define
$U^{\prime}\in G\times H^{\prime}$ by
$U^{\prime}=(U\times\\{0\\})+(\\{0\\}\times J)$. Recalling
$J=\langle(b^{\star},-m)\rangle$ and
$(ma,(ma)^{\star})=(b,b^{\star})\in{\mathcal{L}}$, we then have

$\begin{split}{\mathcal{L}}^{\prime}\cap
U^{\prime}&=(({\mathcal{L}}\times\\{0\\})+(na+x,x^{\star},n)+(\\{0\\}\times
J))\cap((U\times\\{0\\})+(\\{0\\}\times J))\\\
&=(({\mathcal{L}}\times\\{0\\})+\langle m(a,0,1)\rangle+(\\{0\\}\times
J))\cap((U\times\\{0\\})+(\\{0\\}\times J))\\\
&=(({\mathcal{L}}\times\\{0\\})+(\\{0\\}\times J)+(\\{0\\}\times
J))\cap((U\times\\{0\\})+(\\{0\\}\times J))\\\
&=(({\mathcal{L}}\times\\{0\\})+(\\{0\\}\times
J))\cap((U\times\\{0\\})+(\\{0\\}\times J))\\\ &=((({\mathcal{L}}\cap
U)\times\\{0\\})+(\\{0\\}\times J))=\\{0\\}\times J\ .\end{split}$

In the second equality, we used that the ${\mathbb{Z}}$-component is
restricted to be an integer multiple of $m$. In the third equality, we used
$sm(a,0,1)+({\mathcal{L}}\times\\{0\\})=s(0,-b^{\star},m)+({\mathcal{L}}\times\\{0\\})$
for any $s\in{\mathbb{Z}}$. The above proves that ${\mathcal{L}}^{\prime}$ is
a lattice in $G\times H^{\prime}$.

We show that ${\mathcal{L}}^{\prime}$ projects injectively to $G$. Let
$\ell^{\prime}\in{\mathcal{L}}^{\prime}$ such that $\pi^{G}(\ell^{\prime})=0$.
Write $\ell^{\prime}=(na+x,x^{\star},n)+(\\{0\\}\times J)$ for
$(x,x^{\star})\in{\mathcal{L}}$ and $n\in{\mathbb{Z}}$. Then
$0=\pi^{G}(\ell^{\prime})=na+x$. By the remark at the beginning of Section 3.2
we thus have $(na+x,x^{\star},n)\in(0\times J)$, which implies
$\ell^{\prime}=(\\{0\\}\times J)$.

We also have that $\pi^{H^{\prime}}({\mathcal{L}}^{\prime})$ is dense in
$H^{\prime}$. Indeed we have

$\begin{split}\overline{\pi^{H^{\prime}}({\mathcal{L}}^{\prime})}&=\overline{\pi^{H^{\prime}}({\mathcal{L}}+\langle(a,0,1)\rangle+(\\{0\\}\times
J))}=\overline{\pi^{(H\times{\mathbb{Z}})}({\mathcal{L}}+\langle(a,0,1)\rangle)+J}\\\
&\supseteq\overline{\pi^{(H\times{\mathbb{Z}})}({\mathcal{L}}+\langle(a,0,1)\rangle)}+J=(H\times{\mathbb{Z}})+J=H^{\prime}\
,\end{split}$

where we used continuity of the quotient map for the above inclusion. This
proves (a).

“(b)” follows by direct calculation. Let $W^{\prime}=((W\times\\{n\\})+J)\in
H^{\prime}$. Then

$\begin{split}\pi^{G}({\mathcal{L}}&{}^{\prime}\cap(G\times W^{\prime}))\\\
&=\pi^{G}((({\mathcal{L}}\times\\{0\\})+\langle(a,0,1)\rangle)+(\\{0\\}\times
J))\cap(G\times((W\times\\{n\\})+J)))\\\
&=\pi^{G}((({\mathcal{L}}\times\\{0\\})+n(a,0,1)+(\\{0\\}\times
J))\cap(G\times((W\times\\{n\\})+J)))\\\
&=na+\pi^{G}((({\mathcal{L}}\times\\{0\\})+(\\{0\\}\times
J))\cap(G\times((W\times\\{0\\})+J)))\\\
&=na+\pi^{G}({\mathcal{L}}\cap(G\times W))=na+\Lambda_{W}\ .\end{split}$

Here the third equation follows using arguments similar to the above proof
uniform discreteness, and the fifth equation uses the map
$\ell\mapsto\ell\times\\{0\\}+\\{0\\}\times J$, which yields a bijection
between the corresponding subsets of $G\times H$ and $G\times H^{\prime}$.

“(c)” Consider the canonical group homomorphism $f:H\to H^{\prime}$ given by
$h\mapsto(h,0)+J$. Since $\\{0\\}$ is open in ${\mathbb{Z}}$, it is immediate
that the map $H\ni x\mapsto(x,0)\in H\times\\{0\\}$ is a continuous embedding
of $H$ into $H\times{\mathbb{Z}}$. Next, the canonical projection
$\pi:H\times{\mathbb{Z}}\to H\times{\mathbb{Z}}/J$ is a continuous group
homomorphism which is an open mapping [19, p. 59]. Therefore, the map $f$ is a
continuous open group homomorphism. Finally, it follows immediately from the
definition of $J$ that the restriction of $\pi$ to $H\times\\{0\\}$ is one to
one, which gives that $f$ is one to one and hence an embedding.

“(d)” By “(c)”, $f(H)\subseteq H^{\prime}$ is homeomorphic to $H$ and open in
$H^{\prime}$. Thus if $W$ is precompact, then $f(W)$ is precompact in $f(H)$,
which by definition of the trace topology implies that $f(W)$ is precompact in
$H^{\prime}$. Similar arguments apply to the case to $W$ having non-empty
interior and $W$ being topologically regular. Also note that $f(W)$ is
measurable for measurable $W$, as the Borel $\sigma$-algebra on $f(H)$ is
contained in the Borel $\sigma$-algebra on $H^{\prime}$. Finally observe
$\partial(f(W))\subseteq f(\partial W)$, which holds as $f$ is open and closed
(for the latter statement note that $f^{-1}|_{f(H)}$ is continuous). Hence
$m_{H^{\prime}}(\partial f(W))\leq m_{H^{\prime}}(f(\partial W))$, where
$m_{H^{\prime}}$ is a Haar measure on $H^{\prime}$. Noting that $A\mapsto
m_{H^{\prime}}(f(A))$ defines a Haar measure on $H$ as
$m_{H^{\prime}}(f(H))>0$, measure-theoretic regularity of $f(W)$ follows from
measure-theoretic regularity of $W$. ∎

We complete the section by proving Theorem 2.2.

###### Proof of Theorem 2.2.

In the notation from the proof of Proposition 3.2, since $f:H\to H^{\prime}$
is an embedding, we can assume without loss of generality that $H$ is a
subgroup of $H^{\prime}$.

“(a)” and “(b)” follow now from the previous result.

“(c)” follows immediately from the definition of $J$, and “(d)” is obvious.

“(e)” follows from Proposition 3.2 (d).

“(f)” is an immediate consequence of the construction of $H^{\prime}$, but
also follows alternatively from the fact that if $H$ is metrisable, then $H$
ia a metrisable neighborhood of zero in $H^{\prime}$ and hence $H^{\prime}$ is
metrisable. ∎

##  4\. Torus parametrisation and the almost model set hull

We start with the following preparatory lemma, which generalises previous
results [21, 6, 9].

######  Lemma 4.1.

Fix some cut-and-project scheme $(G,H,{\mathcal{L}})$, and take open
$U\subseteq H$ and compact $W\subseteq H$ such that $U\subseteq W$. Consider
any $(U,W)$-almost model set $\Gamma\subseteq G$ . Let $(s_{\alpha})$ be a net
in $G$ such that $(s_{\alpha}+\Gamma)$ converges vaguely to some point set
$\Gamma^{\prime}$ and that $((-s_{\alpha},0)+{\mathcal{L}})$ converges to some
$(s,t)+{\mathcal{L}}\in(G\times H)/{\mathcal{L}}$. Then
$\Lambda_{U+t}\subseteq s+\Gamma^{\prime}\subseteq\Lambda_{W+t}$.

###### Proof.

Note first that if $\Gamma^{\prime}=\varnothing$ then $\Gamma$ is not
relatively dense, and hence neither is $\Lambda_{U}$, which gives that
$U=\varnothing$. Therefore, the inclusions trivially hold. We can thus focus
on the case $\Gamma^{\prime}\neq\varnothing$. Fix some open zero neighborhood
$V$ so that $\Gamma$ is $V$-uniformly discrete, i.e, $(\Gamma+x)\cap V$
contains at most one point, for all $x\in G$. Fix some $a\in\Gamma^{\prime}$.
Since $\Gamma$ is a subset of a model set and thus has finite local
complexity, vague convergence means that for any compact $K\subseteq G$ there
exist a net $(s_{\alpha,K}^{\prime})$ converging to $0$, with
$s_{\alpha,K}^{\prime}\in V$ for all $\alpha$, and some index $\alpha_{K}$
such that
$(s_{\alpha}+s_{\alpha,K}^{\prime}+\Gamma)\cap(K\cup\\{a\\})=\Gamma^{\prime}\cap(K\cup\\{a\\})$
for all $\alpha>\alpha_{K}$, compare [5]. In particular,

$a\in s_{\alpha}+s_{\alpha,K}^{\prime}+\Gamma\Rightarrow
s_{\alpha,K}^{\prime}\in\left(s_{\alpha}+\Gamma-a\right)\cap V\,.$

The $V$-uniform discreteness of $\Gamma$ implies that there exists a unique
element in $\left(s_{\alpha}+\Gamma-a\right)\cap V$ and therefore,
$s_{\alpha,K}^{\prime}$ is independent of $K$. This shows that there exists
some a net $(s_{\alpha}^{\prime})$ converging to $0$, such that, for all
compact $K$ there exists some index $\alpha_{K}$ such that
$(s_{\alpha}+s_{\alpha}^{\prime}+\Gamma)\cap(K\cup\\{a\\})=\Gamma^{\prime}\cap(K\cup\\{a\\})$
for all $\alpha>\alpha_{K}$. In particular we have

$(s_{\alpha}+s_{\alpha}^{\prime}+\Gamma)\cap K=\Gamma^{\prime}\cap K$

for all $\alpha>\alpha_{K}$. Thus, by replacing $s_{\alpha}$ by
$s_{\alpha}+s_{\alpha}^{\prime}$, we may assume without loss of generality
that $(s_{\alpha}+\Gamma)\cap K=\Gamma^{\prime}\cap K$ for all
$\alpha>\alpha_{K}$.

Let us first prove the second inclusion. Fix $x\in\Gamma^{\prime}$ and pick
some compact $K$ such that $x\in\Gamma^{\prime}\cap K$. Then
$x\in(s_{\alpha}+\Gamma)\cap K\subseteq s_{\alpha}+\Lambda_{W}$ for all
$\alpha>\alpha_{K}$. For any $\alpha>\alpha_{K}$ take some
$x_{\alpha}\in\Lambda_{W}$ such that $x=s_{\alpha}+x_{\alpha}$. By compactness
of $W$, there exists some subnet $(x_{\beta})$ of $(x_{\alpha})$, such that
$(x_{\beta}^{\star})$ converges in $H$ to some $y\in W$. Therefore we have by
assumption on $(s_{\alpha})$ that

$(x+s,y+t)+{\mathcal{L}}=\lim_{\alpha}\,(x-s_{\alpha},y)+{\mathcal{L}}=\lim_{\alpha}\,(x_{\alpha},y)+{\mathcal{L}}=\lim_{\beta}\,(0,y-x_{\beta}^{\star})+{\mathcal{L}}=(0,0)+{\mathcal{L}}\
,$

where last claim follows from $\lim_{\beta}\,(y-x_{\beta}^{\star})=0$. This
shows $(x+s,y+t)\in{\mathcal{L}}$. Therefore $x+s\in\pi^{G}({\mathcal{L}})$
and $(x+s)^{\star}=y+t\in W+t$, which proves the second inclusion.

For the first inclusion let $x\in\Lambda_{U+t}$. Then $x^{\star}-t\in U$. Fix
a compact set $K\subseteq G$ such that $x+s$ is an interior point of $K$. Then
$(s_{\alpha}+\Gamma)\cap K=\Gamma^{\prime}\cap K$ for all $\alpha>\alpha_{K}$.
Now pick compact sets $K_{1}\subseteq G$ and $K_{2}\subseteq H$ such that
${\mathcal{L}}+(K_{1}\times K_{2})=G\times H$. For any $\alpha>\alpha_{K}$ we
can then write

$(x-s-
s_{\alpha},x^{\star}-t)=(k^{1}_{\alpha},k^{2}_{\alpha})+(l_{\alpha},l_{\alpha}^{\star})\
,$

where $(k^{1}_{\alpha},k^{2}_{\alpha})\in K_{1}\times K_{2}$ and
$(l_{\alpha},l_{\alpha}^{\star})\in{\mathcal{L}}$. By compactness, there
exists some subnet $((k^{1}_{\beta},k^{2}_{\beta}))$ in $K_{1}\times K_{2}$
that is convergent to some $(k_{1},k_{2})\in K_{1}\times K_{2}$. Now
$((k^{1}_{\beta},k^{2}_{\beta})+{\mathcal{L}})=((x-s-
s_{\beta},x^{\star}-t)+{\mathcal{L}})$ converges to
$(k_{1},k_{2})+{\mathcal{L}}$ and, by assumption, also to
$(x,x^{\star})+{\mathcal{L}}=(0,0)+{\mathcal{L}}$. Thus
$k_{1}\in\pi^{G}({\mathcal{L}})$ and $k_{2}=k_{1}^{\star}$, and we can write

$(x-s-
s_{\beta}+k_{1}-k^{1}_{\beta},x^{\star}-t+k_{2}-k^{2}_{\beta})=(l_{\beta}+k_{1},l_{\beta}^{\star}+k_{1}^{\star})\,.$

Recall that $(x^{\star}-t+k_{2}-k^{2}_{\beta})$ converges to $x^{\star}-t\in
U$ and that $U$ is open. Thus there exists some index $\beta_{0}$ such that
$x^{\star}-t+k_{2}-k^{2}_{\beta}\in U$ for all $\beta>\beta_{0}$. Hence we
have

$x-s+k_{1}-k^{1}_{\beta}\in s_{\beta}+\Lambda_{U}\subseteq s_{\beta}+\Gamma$

for all $\beta>\beta_{0}$. Moreover, since $x-s+k_{1}-k^{1}_{\beta}$ converges
to $x-s$, and since $x-s$ is an interior point of $K$, there exists some
$\beta_{1}>\beta_{0}$ such that

$x-s+k_{1}-k^{1}_{\beta}\in(s_{\beta}+\Gamma)\cap K=\Gamma^{\prime}\cap
K\subseteq\Gamma^{\prime}$

for all $\beta>\beta_{1}$. As $x-s+k_{1}-k^{1}_{\beta}\in\Gamma^{\prime}$
converges to $x-s$ and $\Gamma^{\prime}$ is closed, we get
$x-s\in\Gamma^{\prime}$, as claimed. ∎

The previous lemma has, in conjunction with Proposition 3.2, the following
consequence.

######  Proposition 4.2.

Let $\Lambda\subseteq G$ be a point set and let
$\Gamma\in\mathbb{X}(\Lambda)$.

  * (a)

If $\Lambda$ is a (regular) model set, then $\Gamma$ is a (regular) model set.

  * (b)

If $\Lambda$ is an inter (regular) model set, then $\Gamma$ is an inter
(regular) model set.

  * (c)

If $\Lambda$ is an almost (regular) model set, then $\Gamma$ is an almost
(regular) model set.

###### Proof.

“(a)”: Let $(G,H,{\mathcal{L}})$ be a cut-and-project scheme and let
$W\subseteq H$ be measure-theoretically regular such that
$\Gamma=\Lambda_{W}$. Then there exists $(s,t)\in G\times H$ such that
$\Lambda_{t+W^{\circ}}\subseteq s+\Gamma\subseteq\Lambda_{t+\overline{W}}$. As
$s+\Gamma\subseteq\Lambda_{t+\overline{W}}\subseteq\pi^{G}({\mathcal{L}})$, we
can define

$W^{\prime}=(t+W^{\circ})\cup(s+\Gamma)^{\star}\ .$

Clearly $W^{\prime}$ has interior points by assumption on $W$. Note that
$W^{\prime}$ is measure-theoretically regular, as $(t+W^{\circ})\subseteq
W^{\prime}\subseteq(t+\overline{W})$ implies $\partial W^{\prime}\subseteq
t+\partial W$. We show $s+\Gamma=\Lambda_{W^{\prime}}$, which yields the claim
of the proposition by Theorem 2.2.

The inclusion $s+\Gamma\subseteq\Lambda_{W^{\prime}}$ is obvious. In order to
prove $\Lambda_{W^{\prime}}\subseteq s+\Gamma$, consider
$g\in\Lambda_{W^{\prime}}$ such that $g^{\star}\in W^{\prime}$. If
$g^{\star}\in t+W^{\circ}$, then clearly $g\in\Lambda_{t+W^{\circ}}\subseteq
s+\Gamma$. If $g^{\star}\in(s+\Gamma)^{\star}$, then there exists some
$\gamma\in s+\Gamma$ such that $g^{\star}=\gamma^{\star}$ and hence
$\gamma-g\in\ker(\star)$. Since $s+\Gamma\in\mathbb{X}(\Lambda)$, there exists
a net $(s_{\alpha})$ in $G$ such that $(s_{\alpha}+\Lambda)$ converges to
$s+\Gamma$ in the vague topology. As $\Lambda=\Lambda_{W}$ is
$\ker(\star)$-periodic, so is $s_{\alpha}+\Lambda$ for all $\alpha$. Thus also
the limit $s+\Gamma$ is $\ker(\star)$-periodic. But this implies $g\in
s+\Gamma$, which proves the claim.

“(b)”, “(c)”: By Lemma 4.1, if $\Lambda$ is an inter/almost (regular) model
set then $\Gamma$ is a shifted inter/almost (regular) model set. The claim
follows from Proposition 3.2. ∎

Under slightly stricter assumptions, we can conclude even more. Let us first
prove the following lemma.

######  Lemma 4.3.

Fix some cut-and-project scheme $(G,H,{\mathcal{L}})$, and take some non-empty
open $U\subseteq H$ and compact $W\subseteq H$ such that $U\subseteq W$.
Consider $\Gamma\subseteq G$ such that
$\Lambda_{U}\subseteq\Gamma\subseteq\Lambda_{W}$. If $W\backslash U$ is
nowhere dense, then there exist some $\Gamma^{\prime}\in\mathbb{X}(\Lambda)$
and some $t\in H$ such that $\Gamma^{\prime}=\Lambda_{t+U}=\Lambda_{t+W}$. In
particular, $\Gamma^{\prime}$ is a model set.

###### Proof.

As usual, by eventually replacing $H$ by the subgroup generated by an open
pre-compact set $K$ which contains $W$, we can assume without loss of
generality that $H$ is compactly generated. Thus $H$ can be assumed
$\sigma$-compact without loss of generality. By [10, Prop. 8.3] there exists
some $t\in H$ such that $(t+(W\backslash
U))\cap\pi^{H}({\mathcal{L}})=\varnothing$. Now choose a net
$((s_{\alpha},s_{\alpha}^{\star}))$ in ${\mathcal{L}}$ such that
$s_{\alpha}^{\star}\to t$, which is possible by denseness. By compactness,
$(s_{\alpha}+\Gamma)$ has a subnet $(s_{\beta}+\Gamma)$ convergent to some
$\Gamma^{\prime}\in\mathbb{X}(\Lambda)$. Thus $(s_{\beta}+\Gamma)$ converges
to $\Gamma^{\prime}$, and
$((-s_{\beta},0)+{\mathcal{L}})=((0,s_{\beta}^{\star})+{\mathcal{L}})$
converges to $(0,t)+{\mathcal{L}}$. Therefore Lemma 4.1 yields
$\Lambda_{U+t}\subseteq\Gamma^{\prime}\subseteq\Lambda_{W+t}$. As
$(t+(W\backslash U))\cap\pi^{H}({\mathcal{L}})=\varnothing$, the claim
follows. ∎

Since any set of zero Haar measure is nowhere dense, we have, in conjunction
with Theorem 2.2, the following result.

######  Corollary 4.4.

Let $\Gamma$ be an almost regular model set. Then there exists some
$\Gamma^{\prime}\in\mathbb{X}(\Gamma)$ which is a regular model set. As a
consequence, if $\Gamma$ is an inter regular model set, then there exists some
$\Gamma^{\prime}\in\mathbb{X}(\Gamma)$ which is a regular model set.

We finish the section with a nice consequence for repetitive point sets.
Recall that a point set $\Lambda$ of finite local complexity (such as a subset
of a model set) is called repetitive if the set

$\\{t\in G:(-t+\Lambda)\cap K=\Lambda\cap K\\}$

is relatively dense in $G$, for any compact set $K$. For point sets of finite
local complexity, repetitivity of $\Lambda$ is equivalent to
$\mathbb{X}(\Lambda)$ being minimal, when equipped with its natural
translation action [6, Fact 3]. By combining the results of this section, we
get:

######  Proposition 4.5.

Let $\Gamma\subseteq G$ be a point set of finite local complexity that is
repetitive. Then, the following are equivalent:

  * (i)

$\Gamma$ is a regular model set.

  * (ii)

$\Gamma$ is an inter regular model set.

  * (iii)

$\Gamma$ is an almost regular model set.

###### Proof.

(i)$\Longrightarrow$(ii) $\Longrightarrow$ (iii) is obvious.

(iii)$\Longrightarrow$ (i): By Corollary 4.4, there exists some
$\Gamma^{\prime}\in\mathbb{X}(\Gamma)$ which is a regular model set. By
minimality and Proposition 4.2, we get that
$\Gamma\in\mathbb{X}(\Gamma^{\prime})$ is a regular model set. ∎

##  5\. Second countable internal spaces

Consider a cut-and-project scheme for some almost (regular) model set. Here we
show that the internal space may be assumed second countable. In fact we will
prove a slightly stronger statement, which is summarised as follows.

######  Proposition 5.1.

Let $(G,H,{\mathcal{L}})$ be a cut-and-project scheme. Let $K\subseteq H$ be
compact and let $U\subseteq K$ be measurable. Then there exist a cut-and-
project scheme $(G,H^{\prime},{\mathcal{L}}^{\prime})$ where $H^{\prime}$ is
second countable, a compact set $K^{\prime}\subseteq H^{\prime}$, an open set
$U^{\prime}\subseteq K^{\prime}$ and some positive constant $c$ such that

$\displaystyle c\cdot m_{H}(U)\leq m_{H^{\prime}}(U^{\prime})$
$\displaystyle\leq m_{H^{\prime}}(K^{\prime})\leq c\cdot m_{H}(K)$

and
$\Lambda^{\prime}_{U^{\prime}}\subseteq\Lambda_{U}\subseteq\Lambda_{K}\subseteq\Lambda^{\prime}_{K^{\prime}}$.
In particular, any almost (regular) model set is an almost (regular) model set
in some cut-and-project scheme with second countable internal space.

The proof of Proposition 5.1 will be given below. It is based on the following
lemma.

######  Lemma 5.2.

Let $H$ be a locally compact abelian group with topology $\tau$. Let
$U\subseteq H$ be any zero neighborhood, and let
$\mathcal{D}=\\{f_{i}:i\in\mathbb{N}\\}\subseteq C_{\mathsf{c}}(H)$ be any
countable uniformly bounded family of functions. Then there exist a metrisable
group $(\widetilde{H},\widetilde{d})$ and a continuous group homomorphism
$\phi:(H,\tau)\to(\widetilde{H},\widetilde{d})$ with dense range such that:

  * (a)

For every $f\in\mathcal{D}$ there exists $\widetilde{f}\in
C_{\mathsf{c}}(\widetilde{H})$ such that $f=\widetilde{f}\circ\phi$.

  * (b)

There exists some zero neighborhood $\widetilde{V}\subseteq\widetilde{H}$ such
that $\phi^{-1}(\widetilde{V})\subseteq U$.

###### Proof.

Pick some $f_{0}\in C_{\mathsf{c}}(H)$ such that
$\operatorname{supp}(f_{0})\subseteq U$ and $f_{0}(0)=1$. We then have
$\\{s\in H:\|T_{s}f_{0}-f_{0}\|_{\infty}<1\\}\subseteq U$, where
$(T_{s}f)(h)=f(h+s)$. For $n\in\mathbb{N}_{0}$ define translation invariant
pseudometrics $d_{n}$ on $H$ by
$d_{n}(s,t)=\|T_{s}f_{n}-T_{t}f_{n}\|_{\infty}$. Then the family
$(d_{n})_{n\in\mathbb{N}_{0}}$ is uniformly bounded, since $\mathcal{D}$ is
uniformly bounded. Therefore, we can define a translation invariant
pseudometric $d$ on $H$ via $d=\sum_{n\in\mathbb{N}_{0}}2^{-n}d_{n}$. Then
$Id:(H,\tau)\to(H,d)$ is continuous. Indeed, since any finite set of functions
in $C_{\mathsf{c}}(H)$ is equi-uniformly continuous, for each $\varepsilon>0$
there exists some zero neighborhood $V\subseteq H$ such that
$V\subseteq\\{t\in H:d(t,0)<\varepsilon\\}$. Let
$\phi^{\prime}:(H,d)\to(\widetilde{H},\widetilde{d})$ be the completion map
and define $\phi:(H,\tau)\to(\widetilde{H},\widetilde{d})$ by
$\phi=\phi^{\prime}\circ Id$. By construction $\widetilde{H}$ is a locally
compact abelian group, and $\phi$ is a continuous group homomorphism with
dense range, see e.g. [7, Ch. III, Thm. I, Thm. II]. We verify properties (a)
and (b).

“(a)” Consider arbitrary $n\in\mathbb{N}_{0}$. We show that $f_{n}$ is
uniformly continuous on $(H,d)$. Let $\varepsilon>0$ and define
$\delta=2^{-n}\varepsilon$. If $d(s,t)\leq\delta$ for some $s,t\in H$, we then
have $2^{-n}d_{n}(s,t)\leq\delta$ and hence

$|f_{n}(s)-f_{n}(t)|\leq\|T_{s}f_{n}-T_{t}f_{n}\|_{\infty}=d_{n}(s,t)\leq
2^{n}\delta=\varepsilon\,.$

As $f_{n}$ is uniformly continuous on $(H,d)$, it can be extended uniquely to
the completion, i.e., there exists some $\widetilde{f}_{n}\in
C_{\mathsf{u}}(\widetilde{H})$ such that $\widetilde{f}_{n}\circ\phi=f_{n}$.
We claim that $\widetilde{f}_{n}$ has compact support. Define
$K_{n}=\operatorname{supp}(f_{n})$ and consider the set
$\phi(K_{n})\subseteq\widetilde{H}$, which is compact since $\phi$ is
continuous. It suffices to show that $\widetilde{f}_{n}(\widetilde{x})\neq 0$
implies $\widetilde{x}\in\phi(K_{n})$. Thus consider
$\widetilde{x}\in\widetilde{H}$ such that
$\widetilde{f}_{n}(\widetilde{x})\neq 0$. By denseness of $\phi(H)$ in
$\widetilde{H}$ and by metrisability of $\widetilde{H}$, we can find a
sequence $(x_{k})_{k\in\mathbb{N}}$ in $H$ such that
$\phi(x_{k})\to\widetilde{x}$ as $k\to\infty$. We then have
$\widetilde{f}_{n}(\phi(x_{k}))\to\widetilde{f}_{n}(\widetilde{x})$ as
$k\to\infty$. Since $\widetilde{f}_{n}(\widetilde{x})\neq 0$, there exists
some $k_{0}\in\mathbb{N}$ such that for all $k>k_{0}$ we have
$f_{n}(x_{k})=\widetilde{f}_{n}(\phi(\widetilde{x}_{k}))\neq 0$. This shows
that $x_{k}\in K_{n}$ for all $k>k_{0}$ and hence $\phi(x_{k})\in\phi(K_{n})$
for all $k>k_{0}$. Since $\phi(K_{n})$ is compact and
$\phi(x_{k})\to\widetilde{x}$ as $k\to\infty$, we get
$\widetilde{x}\in\phi(K_{n})$.

“(b)” We show that
$\widetilde{V}=\\{\widetilde{s}\in\widetilde{H}:\widetilde{d}(\widetilde{s},\widetilde{0})<1\\}$
has the desired property. Indeed, let $s\in\phi^{-1}(\widetilde{V})\subseteq
H$. Then $\phi(s)\in\widetilde{V}$ gives
$\|T_{s}f_{0}-f_{0}\|_{\infty}=d_{0}(s,0)\leq
d(s,0)=\widetilde{d}(\phi(s),\widetilde{0})<1$. Thus $s\in U$, which proves
the claim. ∎

######  Remark 5.3 (On injectivity of $\phi$).

From the proof of Lemma 5.2 we infer

$\begin{split}\ker(\phi)&=\\{s\in
H:d(s,0)=0\\}=\bigcap_{n\in\mathbb{N}_{0}}\left\\{s\in
H:d_{n}(s,0)=0\right\\}=\bigcap_{n\in\mathbb{N}_{0}}\text{Per}(f_{n})\
,\end{split}$

where $\text{Per}(f_{n})$ denotes the set of periods of $f_{n}$. In
particular, $\phi$ can be chosen injective if $H$ is metrisable. Indeed take
$\psi\in C_{\mathsf{c}}(H)$ such that $0\leq\psi\leq 1$ and $\psi(0)=1$, which
is possible by Urysohn’s lemma. Take any metric $d$ compatible with the
topology of $H$ and define $\varphi\in C_{\mathsf{c}}(H)$ by
$\varphi(h)=\psi(h)\cdot\left(1-d(h,0)\right)$. Then $\varphi$ is aperiodic
since $\varphi(h)=1$ if and only if $h=0$.

###### Proof of Proposition 5.1.

By regularity of the Haar measure $m_{H}$ on $H$ and by Urysohn’s lemma, we
can find an increasing sequence $(f_{n})_{n\in\mathbb{N}}$ in
$C_{\mathsf{c}}(H)$ and a decreasing sequence $(g_{n})_{n\in\mathbb{N}}$ in
$C_{\mathsf{c}}(H)$ such that $\|g_{n}\|_{\infty}=1$, $m_{H}(g_{n}-1_{K})\leq
1/n$, $m_{H}(1_{U}-f_{n})\leq\frac{1}{n}$ and $0\leq f_{n}\leq 1_{U}\leq
1_{K}\leq g_{n}\leq 1$ for all $n\in\mathbb{N}$. Since ${\mathcal{L}}$ is
lattice, there exists an open zero neighborhood $O_{G}\times O_{H}\subseteq
G\times H$ such that ${\mathcal{L}}\cap(O_{G}\times O_{H})=\\{(0,0)\\}$. We
apply Lemma 5.2 with $\mathcal{D}:=\\{f_{n},g_{n}:n\in\mathbb{N}\\}$ and
$U=O_{H}$. Therefore, there exist a group $\widetilde{H}$ and some group
homomorphism $\phi:H\to\widetilde{H}$ satisfying the conclusion of Lemma 5.2.
Define ${\mathcal{L}}^{\prime}:=\\{(x,\phi(x^{\star})):x\in L\\}$ and
$H^{\prime}=\widetilde{H}$. We split the rest of the proof into steps.

Step 1: We show that $(G,H^{\prime},{\mathcal{L}}^{\prime})$ is a cut-and-
project scheme. First we show that the group ${\mathcal{L}}^{\prime}$ is a
lattice. By Lemma 5.2, there exists an open zero neighborhood
$O_{H^{\prime}}\subseteq H^{\prime}$ such that
$\phi^{-1}(O_{H^{\prime}})\subseteq O_{H}$. It follows immediately that
${\mathcal{L}}^{\prime}\cap(O_{G}\times O_{H^{\prime}})=\\{(0,0)\\}$. Hence
${\mathcal{L}}^{\prime}$ is uniformly discrete. In order to show relative
denseness of ${\mathcal{L}}^{\prime}$, note that there exist compact sets
$K_{G}\subseteq G$ and $K_{H}\subseteq H$ such that
${\mathcal{L}}+(K_{G}\times K_{H})=G\times H$. Then, by continuity of the
completion map we have that $\phi(K_{H})$ is compact and
${\mathcal{L}}^{\prime}+(K_{G}\times\phi(K_{H}))=G\times\phi(H)$. Since
$\phi(H)$ is dense in $H^{\prime}$, there exists a compact sets
$K_{H^{\prime}}$ with non-empty interior such that
$\phi(H)+K_{H^{\prime}}=H^{\prime}$ and hence
${\mathcal{L}}^{\prime}+(K_{G}\times(\phi(K_{H})+K_{H^{\prime}}))=G\times
H^{\prime}$. Hence the group ${\mathcal{L}}^{\prime}$ is a Delone set, i.e.,
${\mathcal{L}}^{\prime}$ is a lattice. The lattice ${\mathcal{L}}^{\prime}$
projects injectively to $G$ as ${\mathcal{L}}$ does and as
$\phi(0^{\star})=0\in H^{\prime}$. In order to show that
${\mathcal{L}}^{\prime}$ projects densely to $H^{\prime}$, let $O$ be any non-
empty open set in $H^{\prime}$. Then $\phi^{-1}(O)\subseteq H$ is open by
continuity of the completion map, and $\phi^{-1}(O)$ is non-empty by denseness
of $H$ in $H^{\prime}$. Therefore there exists some $x\in L$ such that
$x^{\star}\in\phi^{-1}(O)$. But this means $\phi(x^{\star})\in O$.

Step 2: We construct the sets $U^{\prime}$ and $K^{\prime}$. By Lemma 5.2
there exist $f^{\prime}_{n},g^{\prime}_{n}\in C_{\mathsf{c}}(H^{\prime})$ such
that $f_{n}=f^{\prime}_{n}\circ\phi$ and $g_{n}=g^{\prime}_{n}\circ\phi$.
Denseness of $\phi$ combined with $0\leq f_{1}\leq\ldots\leq
f_{n}\leq\ldots\leq g_{n}\leq\ldots\leq g_{1}\leq 1$ gives $0\leq
f^{\prime}_{1}\leq\ldots\leq f^{\prime}_{n}\leq\ldots\leq
g^{\prime}_{n}\leq\ldots\leq g^{\prime}_{1}\leq 1$. Now define

$U^{\prime}=\\{h\in H^{\prime}:f^{\prime}_{n}(h)>0\text{ for some
}n\in\mathbb{N}\\}\ ,\qquad K^{\prime}=\\{h\in
H^{\prime}:g^{\prime}_{n}(h)=1\text{ for all }n\in\mathbb{N}\\}\ .$

By definition $U^{\prime}$ is open in $H^{\prime}$, and $K^{\prime}$ is a
closed subset of $\operatorname{supp}(g^{\prime}_{1})$ and hence compact in
$H^{\prime}$. We show that $U^{\prime}\subseteq K^{\prime}$. Let $h\in
U^{\prime}$. Then, since $\phi(H)$ is dense in $H^{\prime}$ and $U^{\prime}$
is open, there exists a sequence $(x_{k})_{k\in\mathbb{N}}$ in $H$ such that
$\phi(x_{k})\in U^{\prime}$ and $\phi(x_{k})\to h$ as $k\to\infty$. Note that,
by construction, for every $k$ there exists some $n=n(k)$ such that
$0<f^{\prime}_{n}(\phi(x_{k}))=f_{n}(x_{k})\leq 1_{U}(x_{k})$. In particular
this yields $x_{k}\in U\subseteq K$ and hence
$g^{\prime}_{m}(\phi(x_{k}))=g_{m}(x_{k})=1$ for all $m\in\mathbb{N}$. This
gives $\phi(x_{k})\in K^{\prime}$ for all $k\in\mathbb{N}$. As $\phi(x_{k})\to
h$ and $K^{\prime}$ is compact, we get $x\in K^{\prime}$. This proves
$U^{\prime}\subseteq K^{\prime}$.

Step 3: Fix a Haar measure $m_{H^{\prime}}$ on $H^{\prime}$. We prove that
there exists some constant $c\in(0,\infty)$ such that
$m_{H^{\prime}}(K^{\prime})\leq c\cdot m_{H}(K)$ and
$m_{H^{\prime}}(U^{\prime})\geq c\cdot m_{H}(U)$. Consider the pushforward of
the Haar measure $m_{H}$ on $H$ through $\phi$. This is an $H$-invariant
measure on $H^{\prime}$ and hence, by denseness, an $H^{\prime}$-invariant
measure on $H^{\prime}$. Therefore there exists a constant $d\in[0,\infty)$
such that for all $f^{\prime}\in C_{\mathsf{c}}(H^{\prime})$ we have

$\int_{H}f^{\prime}\circ\phi\,\,\mathrm{d}m_{H}=d\cdot\int_{H^{\prime}}f^{\prime}\,\mathrm{d}m_{H^{\prime}}\
.$

Applying this formula to $f^{\prime}=g^{\prime}_{1}$, we get $d\cdot
m_{H^{\prime}}(g^{\prime}_{1})=m_{H}(g^{\prime}_{1}\circ\phi)=m_{H}(g_{1})>0$,
which gives $d\neq 0$. Therefore we have for all $n\in\mathbb{N}$ that

$d\cdot m_{H^{\prime}}(K^{\prime})\leq
d\cdot\int_{H}g_{n}^{\prime}(t)\,\mathrm{d}m_{H}=\int_{H}g_{n}^{\prime}\circ\phi\,\,\mathrm{d}m_{H}=\int_{H}g_{n}\,\,\mathrm{d}m_{H}\,.$

Letting $n\to\infty$ we get $m_{H^{\prime}}(K^{\prime})\leq m_{H}(K)/d$.

Next, $0\leq f_{n}\leq 1_{U}$ gives that $0\leq f_{n}^{\prime}(x^{\prime})\leq
1$ on the dense subset $\phi(H)\subseteq H^{\prime}$ and hence, via
continuity, $0\leq f^{\prime}_{n}(x^{\prime})\leq 1$ for all $n\in\mathbb{N}$
and $x^{\prime}\in H^{\prime}$. The definition of $U^{\prime}$ then gives
$0\leq f_{n}^{\prime}\leq 1_{U^{\prime}}$. Therefore

$d\cdot m_{H^{\prime}}(U^{\prime})\geq
d\cdot\int_{H}f_{n}^{\prime}(t)\,\mathrm{d}m_{H}=\int_{H}f_{n}^{\prime}\circ\phi\,\,\mathrm{d}m_{H}=\int_{H}f_{n}\,\,\mathrm{d}m_{H}\,.$

Letting $n\to\infty$ we get $m_{H^{\prime}}(U^{\prime})\geq m_{H}(U)/d$. The
claim now follows with $c=1/d$.

Step 4: We prove $\Lambda^{\prime}_{U^{\prime}}\subseteq\Lambda_{U}$ and
$\Lambda_{K}\subseteq\Lambda^{\prime}_{K^{\prime}}$. Let
$x\in\Lambda^{\prime}_{U^{\prime}}$. Then $\phi(x^{\star})\in U^{\prime}$, and
hence there exists some $n\in\mathbb{N}$ such that
$f_{n}(x^{\star})=f^{\prime}_{n}(\phi(x^{\star}))>0$. Since $f_{n}\leq 1_{U}$
we get that $x^{\star}\in U$ and hence $x\in\Lambda_{U}$. This shows
$\Lambda^{\prime}_{U^{\prime}}\subseteq\Lambda_{U}$. If $x\in\Lambda_{K}$ we
have $x^{\star}\in K$ and hence for all $n\in\mathbb{N}$ we have
$g^{\prime}_{n}(\phi(x^{\star}))=g_{n}(x^{\star})=1$. This means
$\phi(x^{\star})\in K^{\prime}$ and $x^{\star}\in\Lambda_{K^{\prime}}$.

It follows immediately that any almost model set
$\Lambda_{U}\subseteq\Gamma\subseteq\Lambda_{K}$ in $(G,H,{\mathcal{L}})$
satisfies $\Lambda_{U^{\prime}}\subseteq\Gamma\subseteq\Lambda_{K^{\prime}}$
and hence is an almost model set in $(G,H^{\prime},{\mathcal{L}}^{\prime})$.
If furthermore $m_{H}(K\backslash U)=0$, then step 3 gives
$m_{H^{\prime}}(K^{\prime}\backslash U^{\prime})=0$, and hence measure-
theoretic regularity is preserved.

At this point we have proved the proposition – apart from $\sigma$-compactness
of $H^{\prime}$. But $\sigma$-compactness may be assumed without loss of
generality, as $K^{\prime}$ lies inside a compactly generated open and closed
subgroup of $H^{\prime}$. ∎

##  6\. Extended cut-and-project schemes

In this section we restrict to cut-and-project schemes
$({\mathbb{R}}^{d},H,{\mathcal{L}})$ with Euclidean direct space and second
countable internal space $H$. Thus ${\mathcal{L}}$ is countable, and also the
annihilator ${\mathcal{L}}^{\circ}\subseteq{\mathbb{R}}^{d}\times\widehat{H}$
of ${\mathcal{L}}$ is countable.

######  Lemma 6.1.

Let $A$ be any countable subset of ${\mathbb{R}}^{d}$. Then there exists a
lattice $D$ in ${\mathbb{R}}^{d}$ such that
$D\cap\mathrm{Span}_{{\mathbb{Q}}}(A)=\\{0\\}$ and
$D^{\circ}\cap\mathrm{Span}_{{\mathbb{Q}}}(A)=\\{0\\}$, where
$D^{\circ}=\\{x\in{\mathbb{R}}^{n}:\langle d,x\rangle\in{\mathbb{Z}}\text{ for
all }d\in D\\}$ is the lattice dual to $D$.

###### Proof.

We will use the coordinate projections
$\pi_{i}:{\mathbb{R}}^{d}\to{\mathbb{R}}$, given by
$(x_{1},\ldots,x_{d})^{T}\mapsto x_{i}$. Define the set
$S=\bigcup_{i=1}^{d}\pi_{i}(A)\subseteq{\mathbb{R}}$ of all entries of vectors
from $A$, and consider the ${\mathbb{Q}}$-vector space
$V=\mathrm{Span}_{{\mathbb{Q}}}(S)\subseteq{\mathbb{R}}$. Let $T$ be a
countable ${\mathbb{Q}}$-basis for $V$. We claim that there exist nonzero real
numbers $c_{1},\ldots,c_{d}$ such that the set

$T^{\prime}=T\cup\\{c_{1},\ldots,c_{d}\\}\cup\\{c_{1}^{-1},\ldots,c_{d}^{-1}\\}$

is ${\mathbb{Q}}$-linearly independent. This is seen inductively. Indeed, if
$T\cup\\{c,\frac{1}{c}\\}$ is ${\mathbb{Q}}$-linearly dependent, then $c$
satisfies a quadratic equation over the countable multivariate polynomial ring
${\mathbb{Q}}[T]$. As the number of quadratic equations with coefficients in
${\mathbb{Q}}[T]$ is countable, hence so is the number of roots of such
equations. Since ${\mathbb{R}}$ is uncountable, we can pick a
$c\in{\mathbb{R}}$ which is not the root of such an equation.

Consider now the lattice
$D=\mathrm{Span}_{{\mathbb{Z}}}(\\{c_{1}e_{1},\ldots,c_{d}e_{d}\\})\subseteq{\mathbb{R}}^{d}$,
where $e_{i}\in{\mathbb{R}}^{d}$ is the $i^{th}$ coordinate vector, i.e., we
have $\pi_{j}(e_{i})=\delta_{ij}$ for $i,j\in\\{1,\ldots,d\\}$. In order to
show $D\cap\mathrm{Span}_{{\mathbb{Q}}}(A)=\\{0\\}$, consider any $u\in
D\cap\mathrm{Span}_{{\mathbb{Q}}}(A)$. Then there exist vectors
$a_{1},\ldots,a_{k}\in A$, rationals $q_{1},\ldots,q_{k}\in{\mathbb{Q}}$ and
integers $n_{1},\ldots,n_{d}\in{\mathbb{Z}}$ such that

$u=\sum_{i=1}^{d}n_{i}c_{i}e_{i}=\sum_{j=1}^{k}q_{j}a_{j}\ .$

For the vector components we have
$n_{i}c_{i}=\sum_{j=1}^{k}q_{j}\pi_{i}(a_{j})$ for $i\in\\{1,\ldots,d\\}$.
Using ${\mathbb{Q}}$-linear independence of $T^{\prime}$ we conclude
$n_{1}=\ldots=n_{d}=0$ and $q_{1}=\ldots=q_{k}=0$, which proves
$D\cap\mathrm{Span}_{{\mathbb{Q}}}(A)=\\{0\\}$. The claim
$D^{\circ}\cap\mathrm{Span}_{{\mathbb{Q}}}(A)=\\{0\\}$ is shown analogously,
noting
$D^{\circ}=\mathrm{Span}_{{\mathbb{Z}}}(\\{c_{1}^{-1}e_{1},\ldots,c_{d}^{-1}e_{d}\\})\subseteq{\mathbb{R}}^{d}$.
∎

######  Lemma 6.2.

Let $({\mathbb{R}}^{d},H,{\mathcal{L}})$ be a cut-and-project scheme such that
${\mathcal{L}}$ and ${\mathcal{L}}^{\circ}$ are both countable. Let $D$ be a
lattice in ${\mathbb{R}}^{d}$ such that $D\cap\pi^{G}({\mathcal{L}})=\\{0\\}$
and $D^{\circ}\cap\pi^{\widehat{G}}({\mathcal{L}}^{\circ})=\\{0\\}$. Consider
the compact abelian group $\mathbb{K}={\mathbb{R}}^{d}/D$ with quotient map
$\psi:G\to\mathbb{K}$. Then the following hold.

  * (a)

$\psi$ is one-to-one on $\pi^{G}({\mathcal{L}})$, and
$\psi(\pi^{G}({\mathcal{L}}))$ is dense in $\mathbb{K}$.

  * (b)

For each non-empty open set $U\subseteq H$ the set $\psi(\Lambda_{U})$ is
dense in $\mathbb{K}$.

######  Remark 6.3.

As the following proof shows, the construction in Lemma 6.2 works more
generally for cut-and-project schemes with second countable $G$ and $H$, if
there is a continuous group homomorphism to a compact metrisable group
$\mathbb{K}$, such that $\ker(\psi)\cap\pi^{G}({\mathcal{L}})=\\{0\\}$ and
$\ker(\widehat{\psi})\cap\pi^{\widehat{G}}({\mathcal{L}}^{0})=\\{0\\}$, where
$\widehat{\psi}$ denotes the group homomorphism dual to $\psi$. In the setting
of the lemma, the latter condition may be satisfied due to Lemma 6.1.

###### Proof.

Existence of a lattice $D\subset{\mathbb{R}}^{d}$ such that
$D\cap\pi^{G}({\mathcal{L}})=\\{0\\}$ and
$D^{\circ}\cap\pi^{\widehat{G}}({\mathcal{L}}^{\circ})=\\{0\\}$ follows from
Lemma 6.1.

(a) Injectivity of $\psi|_{\pi^{G}({\mathcal{L}})}$ is immediate from
$\pi^{G}({\mathcal{L}})\cap D=\\{0\\}$. Denseness is a consequence of
Pontryagin duality, compare [18, Prop. 4.1]: The map
$\alpha:{\mathcal{L}}\to\mathbb{K}$, given by
$\ell\mapsto\psi(\pi^{G}(\ell))$, is a continuous group homomorphism. Thus
$\psi(\pi^{G}({\mathcal{L}}))$ is dense in $\mathbb{K}$ if the dual group
homomorphism
$\widehat{\alpha}:D^{\circ}\to(\widehat{G}\times\widehat{H})/{\mathcal{L}}^{\circ}$,
given by $\chi\mapsto(\chi,1)+{\mathcal{L}}^{\circ}$, is one-to-one. Here
$1\in\widehat{H}$ denotes the trivial character. But the latter holds as
$D^{\circ}\cap\pi^{\widehat{G}}({\mathcal{L}}^{\circ})=\\{0\\}$.

(b) Here we follow ideas behind Weyl’s uniform distribution theorem. Consider
the space $\mathcal{M}_{f}(\mathbb{K})$ of finite Borel measures on
$\mathbb{K}$, equipped with the topology of weak convergence. Fix the van Hove
sequence $(A_{n})_{n\in\mathbb{N}}$ in ${\mathbb{R}}^{d}$ where
$A_{n}=[-n,n]^{d}$, and let $V\subseteq U\subseteq H$ be a compact measure-
theoretically regular set having non-empty interior. Consider measures
$\mu_{n}\in\mathcal{M}_{f}(\mathbb{K})$ given by

$\mu_{n}=\frac{1}{m_{G}(A_{n})}\sum_{x\in
A_{n}\cap\Lambda_{V}}\delta_{\psi(x)}\ ,$

where $m_{G}$ is the Lebesgue measure. Then $(\mu_{n})_{n\in\mathbb{N}}$ is a
sequence of equi-bounded measures in $\mathcal{M}_{f}(\mathbb{K})$. We claim
that $(\mu_{n})_{n\in\mathbb{N}}$ converges weakly to
$\operatorname{dens}(\Lambda_{V})\cdot m_{\mathbb{K}}$, where $m_{\mathbb{K}}$
is the Haar probability measure on $\mathbb{K}$.

Since $\mathbb{K}$ is metrisable, any set of equi-bounded measures in
$\mathcal{M}_{f}(\mathbb{K})$ is compact and metrisable. Therefore it suffices
to show that any convergent subsequence converges vaguely to
$\operatorname{dens}(\Lambda_{V})\cdot m_{\mathbb{K}}$. Let
$(\mu_{k_{n}})_{n\in\mathbb{N}}$ be a subsequence of
$(\mu_{n})_{n\in\mathbb{N}}$ that converges vaguely to $\mu$. Consider the
Fourier-Stieltjes transform $\widehat{\mu}$ of $\mu$, which is a uniformly
continuous and bounded function on $\widehat{\mathbb{K}}\cong D^{\circ}$.
Then, for all $\chi\in D^{0}$ we have by weak convergence and $D$-invariance
that

$\widehat{\mu}(\chi)=\int_{\mathbb{K}}\chi(t)\,\mathrm{d}\mu(t)=\lim_{n}\int_{\mathbb{K}}\chi(t)\,\mathrm{d}\mu_{k_{n}}(t)=a_{\chi}(\delta_{\Lambda_{V}})\
,$

where
$a_{\chi}(\nu)=\lim_{n\to\infty}\frac{1}{|A_{k_{n}}|}\int\overline{\chi}(t){\rm
d}\nu(t)$ denotes the Fourier-Bohr coefficient of $\nu$ at $\chi$ along
$(A_{k_{n}})_{n}$. Existence and properties of Fourier-Bohr coefficients for
regular model sets have been discussed e.g. in [16, Sec. 7] and in [17, Cor.
3.40]. For $\chi=1$ we have
$a_{1}(\delta_{\Lambda_{V}})=\operatorname{dens}(\Lambda_{V})$. By
construction, for $\chi\neq 1$ we have
$\chi\notin\pi^{\widehat{G}}({\mathcal{L}}^{\circ})$ and hence
$a_{\chi}(\delta_{\Lambda_{V}})=0$. Obviously, the measure
$\mu^{\prime}=\operatorname{dens}(\Lambda_{V})\cdot m_{\mathbb{K}}$ has the
same Fourier-Stieltjes transform as $\mu$. By injectivity of the Fourier
transform [20, 1.7.3 (b)] we conclude $\mu=\mu^{\prime}$.

We finally show that $\psi(\Lambda_{V})$ is dense in $\mathbb{K}$. Consider an
arbitrary non-empty open set $O\subseteq\mathbb{K}$. Take $\phi\in
C_{\mathsf{c}}(\mathbb{K})$ such that $\operatorname{supp}(\phi)\subseteq O$
and $m_{\mathbb{K}}(\phi)\neq 0$. We then have
$0\neq\mu(\phi)=\lim_{n\to\infty}\mu_{n}(\phi)$. Take some $n$ such that
$0\neq\mu_{n}(\phi)$. Then there exists some
$t\in\operatorname{supp}(\phi)\cap\psi(\Lambda_{V}\cap A_{n})\subseteq
O\cap\psi(\Lambda_{V})$. ∎

######  Lemma 6.4.

Under the conditions of Lemma 6.2, the map $\phi:\pi^{G}({\mathcal{L}})\to
H\times\mathbb{K}$, given by $\phi(x)=(x^{\star},\psi(x))$, has dense range.

###### Proof.

Let $\varnothing\neq U\subseteq H\times\mathbb{K}$ be any open set. Then,
there exist open sets $V\subseteq H$ and $O\subseteq\mathbb{K}$ such that
$\varnothing\neq V\times O\subseteq U$. By Lemma 6.2, the set
$\psi(\Lambda_{V})$ is dense in $\mathbb{K}$. Therefore, there exists some
$x\in\Lambda_{V}$ such that $\psi(x)\in O$. Thus
$\phi(x)=(x^{\star},\psi(x))\in V\times O\subseteq U$. ∎

We can now define an extended cut-and-project scheme with injective
$\star$-map.

######  Theorem 6.5 (Extended cut-and-project scheme).

Let $(G,H,{\mathcal{L}})$ be a cut-and-project scheme with
$G={\mathbb{R}}^{d}$ such that ${\mathcal{L}}$ and ${\mathcal{L}}^{\circ}$ are
both countable. Let $\mathbb{K}$ be as in Lemma 6.2 and define
$H^{\prime}=H\times\mathbb{K}$. Let $\phi:\pi^{G}({\mathcal{L}})\to
H^{\prime}$ be as in Lemma 6.4 and define
${\mathcal{L}}^{\prime}=\\{(x,\phi(x)):x\in\pi^{G}({\mathcal{L}})\\}$. Then
the following hold.

  * (a)

$(G,H^{\prime},{\mathcal{L}}^{\prime})$ is a cut-and-project scheme with
injective $\star$-map $\phi$.

  * (b)

For any $W\subseteq H$ we have $\Lambda_{W}=\Lambda^{\prime}_{W^{\prime}}$
where $W^{\prime}=W\times\mathbb{K}$.

  * (c)

If $H$ is $\sigma$-compact or metrisable, respectively, then so is
$H^{\prime}$.

######  Remark 6.6.

Note that, under the above assumptions, any model set in $(G,H,{\mathcal{L}})$
is a model set in $(G,H^{\prime},{\mathcal{L}}^{\prime})$, and the window
associated to the extended cut-and-project scheme inherits measurability and
any property in $(T)$ or $(M)$.

###### Proof.

“(a)” We show that the subgroup ${\mathcal{L}}^{\prime}$ of $G\times
H^{\prime}$ is a lattice. Since ${\mathcal{L}}$ is a lattice in
$G\times\mathbb{K}$, there exists open sets $0\in U\subseteq G$ and $0\in
V\subseteq H$ such that ${\mathcal{L}}\cap(U\times V)=\\{(0,0)\\}$. Then
${\mathcal{L}}^{\prime}\cap(U\times V\times\mathbb{K})=\\{(0,0,0)\\}$. Hence
${\mathcal{L}}^{\prime}$ is discrete. As to relative denseness, let
$K\subseteq G$ and $W\subseteq H$ be compact sets so that
${\mathcal{L}}+(K\times W)=G\times H$. Then ${\mathcal{L}}^{\prime}+(K\times
W\times\mathbb{K})=G\times H\times\mathbb{K}$. Turning to the projection
assumptions, injectivity of $\pi^{G}|_{{\mathcal{L}}^{\prime}}$ readily
follows from injectivity of $\pi^{G}|_{\mathcal{L}}$. Finally,
$\pi^{H^{\prime}}({\mathcal{L}}^{\prime})$ is dense in $H^{\prime}$ by Lemma
6.4. This proves (a).

“(b)” is obvious as
$\Lambda^{\prime}_{W^{\prime}}=\pi^{G}({\mathcal{L}}^{\prime}\cap(G\times
W\times\mathbb{K}))=\pi^{G}({\mathcal{L}}\cap(G\times W))=\Lambda_{W}$.

“(c)” follows immediately from the fact that the construction in Lemma 6.1
produces a compact and metrisable $\mathbb{K}$ . ∎

###### Proof of Theorem 2.9.

Let $\Lambda\subseteq{\mathbb{R}}^{d}$ be any shifted almost model set. Then
$\Lambda$ is an almost model set by Theorem 2.2, and hence it is an almost
model set in a cut-and-project scheme with second countable internal space by
Proposition 5.1. Applying Theorem 6.5, we get that there exist a cut-and-
project scheme $({\mathbb{R}}^{d},H^{\prime},{\mathcal{L}}^{\prime})$ with
second countable $H^{\prime}$ and injective $\star$-map, and
$U^{\prime}\subseteq K^{\prime}$ such that $U^{\prime}$ is non-empty open,
$K^{\prime}$ is compact and
$\Lambda_{U^{\prime}}\subseteq\Lambda\subseteq\Lambda_{K^{\prime}}$.
Measurability and properties $(T)$ and $(M)$ are inherited due to Remark 6.6.
Furthermore, if $\Lambda$ is an almost regular model set, we also have
$m_{H^{\prime}}(K^{\prime}\backslash U^{\prime})=0$. Defining
$W=U^{\prime}\cup(\Lambda^{\star})$ exactly as in Proposition 2.4, we get
$\Lambda_{W}=\Lambda$, and the rest of the claim follows. The last statement
follows from the observation that any inter (regular) model set is an almost
(regular) model set by Proposition 2.6. ∎

## Acknowledgements

NS was supported by the Natural Sciences and Engineering Council of Canada
(NSERC) via grant 2020-00038, and he is grateful for the support. Part of this
work was done during the ”Summer School on Aperiodic Order” at MacEwan
University which was sponsored by the Research Office, the Dean Office and the
Mathematics and Statistics Department at MacEwan University, as well as
Collaborative Research Centre 1283 of the DFG at the Bielefeld University, and
the authors are grateful for the support.

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