1 Introduction
The distance structure of lattices and periodic point sets is efficiently summarised in the corresponding theta series, which are well studied and understood, and have close connections with modular forms [Reference Conway and Sloane22]. Much less is known for aperiodic point sets, even for the most popular ones, such as the Fibonacci or the silver mean chain. It is questionable to what extent individual theta series are meaningful, as no two members of the Fibonacci hull would have the same series, and similarly for all other (regular) model sets.
A natural question would then be what the averaged distance structure is, where the average is taken over all elements of the hull with a point at 0. This average is well defined and can often be determined, at least in principle, from one element of the hull. For regular model sets, which are particularly nice cut-and-project sets, the answer is then given in terms of (sums of) pair-correlation functions. The latter, in turn, lead to covariograms of the coding windows, or cross covariograms between different windows. These are always continuous functions in internal space, which sounds nice and easy. If the windows are simple polygons, one indeed obtains decent functions. Nevertheless, no clear relation to aperiodic analogues of theta series has materialised so far, despite several efforts in this direction [Reference Alfes, Kiefer and Mazáč3, Reference Baake and Grimm11, Reference Björklund, Hartnick and Karasik21, Reference Moody and Weiss34].
In view of this, potential insight might come from investigating some less prominent examples that are nevertheless somehow typical. Indeed, even within the class of one-dimensional inflation tilings, windows of Rauzy fractals are ubiquitous, but have not been studied from this point of view. One reason certainly is the obstacle of computing the covariogram of a Rauzy fractal, which already in one dimension tends to be a Cantorval [Reference Baake, Gorodetski and Mazáč10]. The covariograms are continuous functions (being convolutions of two functions that are both
$L^{1}$
and
$L^{\infty }$
), with a behaviour that is somewhat reminiscent to that of a Weierstrass monster.
In fact, until recently, hardly any reliable technique for computing these functions was known. Within the class of inflation point sets, the identification of an exact renormalisation scheme for the pair-correlation functions in [Reference Baake, Gähler and Mañibo9, Reference Mañibo29] opens the door to some efficient and exact calculations, as we demonstrate by way of two examples (for more, see [Reference Klick26]). The crucial connection comes from the uniform distribution property of the model set points within the windows, after their lift via the
$\star $
-map. Consequently, the renormalisation scheme permits the calculation of the (cross) covariograms at a dense set of points, which is enough for the illustration of a continuous function.
This looks like a maybe tedious, but ultimately easy thing to do, until one actually does it. Indeed, the spiky behaviour of the covariograms turns out to be unexpectedly wild, and the convergence of the approximants is surprisingly slow, at least in some examples. It is thus the goal of this paper to introduce the problem and method via some characteristic examples, and to demonstrate that this is a fractal phenomenon worth further study.
This paper is organised as follows. In Section 2, we set the scene and introduce the key concepts and notions around covariograms, cut-and-project sets and pair-correlation functions. In the main Section 3, we then derive the results for two binary inflation systems. The first is based on the silver mean inflation factor,
$\lambda = 1+\sqrt {2}$
, and the second on its square. A central block will be the explicit derivation of the exact renormalisation equations, and how they are used to approach the (cross) covariograms.
2 Preliminaries
Let us begin by recalling some concepts and notions; see [Reference Baake and Grimm13] for general background.
Definition 2.1. Let
$W \subset \mathbb {R}^{m}$
be a nonempty compact set. We define the covariogram of W for almost all
$x \in \mathbb {R}^{m}$
as
$$ \begin{align*} \mathrm{cvg}_{W}(x)\, \mathrel{\mathop:}= \, {\mathrm{vol}}(W \cap (x+W)) = ( \mathbf{1}_{-W} \ast \mathbf{1}_{W})(x), \end{align*} $$
where
$\mathbf {1}_{W}$
is the characteristic function of W, and
$\ast $
stands for the usual convolution of two functions
$f, g\in L^{1}(\mathbb {R}^m)$
as given by
$$ \begin{align*} (f\ast g)(x) \, = \int_{\mathbb{R}^m} f(y) g(x-y) \mathop{}\!{d}y, \end{align*} $$
with the usual understanding that this is well defined for almost all
$x\in \mathbb {R}^m$
.
Arguably, this is most famously known in relation to the covariogram problem: whether the covariogram of a convex body in
$\mathbb {R}^m$
determines the body uniquely, up to translations and reflections. The answer is positive for convex polytopes when
$m \leqslant 3$
; see [Reference Bianchi, Koldobsky and Volbergs20]. Variations of this problem exist and apply to our setting, such as determining when two model sets share the same covariogram and the corresponding inverse problem [Reference Baake and Grimm12].
Before turning our attention to the realm of model sets, we briefly recall and collect various properties of a covariogram that immediately follow from its definition, some usual inequalities, and elementary change of variable arguments; see [Reference Baake and Grimm13, Reference Klick26] for further references.
Proposition 2.2. Let
$W, W' \subset \mathbb {R}^{m}$
be nonempty compact sets. Then,
-
(i)
$\mathrm {cvg}_{W} \in C_{\mathtt {c}}(\mathbb {R}^m)$
, and
$\mathrm {cvg}_{W}$
is an even function; -
(ii)
$\mathrm {supp} ( \mathrm {cvg}_{W}) \subseteq W \! - W \mathrel {\mathop :}= \{x-y : x,y\in W \}$
; -
(iii)
$W \hspace {-0.5pt}\subseteq W'$
implies
$\mathrm {cvg}_{W}(x) \, \leqslant \, \mathrm {cvg}_{W'}(x)$
for all
$\in \mathbb {R}^m$
; -
(iv)
$\mathrm {cvg}_{W}$
is Fourier transformable, and one has, for all
$x\in \mathbb {R}^m$
,
$$ \begin{align*} \widehat{\mathrm{cvg}_{W}}(x) = | \widehat{\mathbf{1}_{W}}(x) |^2; \end{align*} $$
-
(v) for all
$b\in \mathbb {R}^m$
and all
$\lambda \in \mathbb {R} $
with
$\lambda> 0$
, one has
$$ \begin{align*} \mathrm{cvg}_{W + b}(x) = \mathrm{cvg}_{W}(x) \quad \text{and} \quad \, \mathrm{cvg}_{\lambda W} (x) = \lambda\, \mathrm{cvg}_{W} \,\bigg( \frac{x}{\lambda} \bigg); \end{align*} $$
-
(vi) if
$\mathrm {int}(W) \cap \mathrm {int}(\mathrm {W'}) = \varnothing $
with
${\mathrm {vol}}(\partial W) = {\mathrm {vol}}(\partial W')=0$
, one has for almost every
$$ \begin{align*} \mathrm{cvg}_{W \cup W'}(x) = \mathrm{cvg}_{W}(x) + \mathrm{cvg}_{W'}(x) + (\mathbf{1}_{-W}\ast \mathbf{1}_{W'}) (x) + (\mathbf{1}_{-W}\ast \mathbf{1}_{W'}) (-x) \end{align*} $$
$x \in \mathbb {R}^m $
.
Clearly, the complexity of determining the covariogram of a given set W depends on its structure. We are interested in finding the covariograms of so-called Rauzy fractals. They are solutions of an iterated function system, and are intimately related to Pisot substitutions and their description as model sets. We briefly summarise this connection in what follows, and refer the reader to [Reference Baake and Grimm13] for further details and to [Reference Meyer32, Reference Siegel and Thuswaldner35, Reference Sing36] for deeper connections.
Let
$\mathcal {A}$
be a finite set of n symbols, called an alphabet. A substitution rule
$\varrho $
is a map that sends each element of
$\mathcal {A}$
(a letter) to a nonempty concatenation of finitely many letters (a word), and extends to all (bi-infinite) words by the endomorphism property. To every substitution rule
$\varrho $
, one can assign the substitution matrix
$M_{\varrho } \in \mathrm {Mat}(n,\, \mathbb {Z})$
with entries
$$ \begin{align*} ( M_{\varrho})_{ij} = \, \text{number of letters of type } i \text{ in } \varrho(j), \end{align*} $$
where we have fixed a numbering of the letters of
$\mathcal {A}$
. A substitution
$\varrho $
is primitive if its substitution matrix
$M_{\varrho }$
is primitive. By iterating a primitive substitution rule, starting with a letter from the alphabet (or, more generally, with a legal word [Reference Baake and Grimm13, Definition 4.5]), we obtain, for (a suitable power of) the substitution, a (bi)-infinite fixed point (a sequence of letters from the alphabet that is invariant under the action of (the power of) the substitution rule).
Since the substitution matrix is a nonnegative matrix, via standard Perron–Frobenius (PF) arguments [Reference Baake and Grimm13, Ch. 2.4], it provides insight into various properties of
$\varrho $
. In particular, if a primitive substitution is additionally a Pisot substitution, meaning the PF eigenvalue of
$M_{\varrho }$
is a Pisot–Vijayaraghavan (PV) number and the characteristic polynomial is irreducible, the substitution leads to a tiling of the real line by assigning a closed interval to each letter whose (natural tile) length is proportional to an entry of the left PF eigenvector of
$M_{\varrho }$
. These intervals are called prototiles. Then, the PF eigenvalue of
$M_{\varrho }$
gives the inflation factor
$\lambda $
of the induced (geometric) inflation rule. In this setting, it is natural to work with the displacement matrix
$T_{\varrho }$
, which is a set-valued analogue of the substitution matrix with entries
$$ \begin{align*} (T_{\varrho})_{ij} = T_{ij} = \{\text{relative positions of tiles of type } i \text{ in supertile } \varrho(j) \}, \end{align*} $$
and one has
$(M_{\varrho })_{ij} = {\mathrm {card}}(T_{ij})$
. Placing a control point on the left endpoint of each tile, we can understand the fixed point of a given substitution as a collection of point sets assigned to each letter. If we collect all control points of tiles of type i, we obtain a Delone set
$\varLambda _{i}\subset \mathbb {R}$
. The set of all control points is denoted by
$\varLambda $
. It satisfies
$$ \begin{align*} \varLambda \, = \bigcup_{k=1}^n \varLambda_{k} \end{align*} $$
and the union is disjoint. Since one considers a tiling of the real line described as a fixed point of a Pisot substitution (inflation) rule
$\varrho $
with n letters (prototiles) with the inflation factor
$\lambda $
, this is reflected on the level of the control points as
$$ \begin{align} \varLambda_i = \bigcup_{j=1}^{n} \, \lambda \varLambda+ T_{ij} = \bigcup_{j=1}^{n} \, \bigcup_{t \in T_{ij}} \lambda \varLambda_j + t , \end{align} $$
where, again, the unions are disjoint. Note that the right-hand side (RHS), viewed as a mapping between n-tuples of Delone sets, does not define a contraction with respect to any natural metric. Consequently, finding a characterisation of the solution space of Equation (2-1) is difficult. We are not aware of any general result in this direction.
If the alphabet is binary and
$\lambda $
is a PV number, the sets
$\varLambda _{i}$
are regular model sets [Reference Hollander and Solomyak23]. For larger alphabets, this problem is known under the name Pisot substitution conjecture. It is unclear whether or not this conjecture should hold, but all known examples with irreducible characteristic polynomial of the substitution matrix satisfy it. We refer the reader to [Reference Akiyama, Barge, Berthé, Lee, Siegel, Kellendonk, Lenz and Savinien1] for a summary on the Pisot substitution conjecture, to [Reference Barge17] for various families for which the conjecture has been verified and to [Reference Balchin and Rust16] for an extensive computer search for a counterexample.
Model sets are special cases of cut-and-project sets with nice properties, both arising from cut-and-project schemes; compare [Reference Meyer32, Reference Moody and Moody33].
Definition 2.3. A cut-and-project scheme (CPS) is a triple
$(\mathbb {R}^d,\, H,\, \mathcal {L})$
consisting of
$\mathbb {R}^d$
(direct/physical space) and a locally compact Abelian group (LCAG) H (internal space), together with a discrete co-compact subgroup, a lattice,
$\mathcal {L} \subset \mathbb {R}^d \times H$
and the natural projections
$\pi : \mathbb {R}^{d} \times H\to \mathbb {R}^{d}$
and
$\pi _{\text {int}} :\mathbb {R}^{d} \times H\to H$
enjoying the following properties:
-
(1) The restriction
$\pi |_{\mathcal {L}}$
is one-to-one. -
(2) The image
$\pi _{\text {int}}(\mathcal {L})$
is dense in H.
Given a CPS
$(\mathbb {R}^d,\, H,\, \mathcal {L})$
, we set
$L= \pi ( \mathcal {L})$
, which is a subgroup of
$\mathbb {R}^d$
. The first condition in the definition of a CPS implies the existence of a mapping
$\star : L \to H$
, called the
$\star $
-map (known as the star map), defined by
$\star = \pi _{\text {int}} \circ ( \pi |_{\mathcal {L}})^{-1}$
. If
$H = \mathbb {R}^m$
, the CPS is called Euclidean. We summarise a CPS via the diagram
as usual. With the
$\star $
-map, we can now define a model set. We refer the reader to [Reference Baake and Grimm13, Section 7] and [Reference Moody and Moody33] for further discussions of model sets and their properties.
Definition 2.4. Given a CPS
$(\mathbb {R}^d,\, H,\, \mathcal {L})$
and an arbitrary, relatively compact set
$W \subset H$
, we denote by
${\Large \curlywedge }(W)$
the set
$$ \begin{align*} {\Large \curlywedge}(W) \, \mathrel{\mathop:}= \, \{ x \in L : x^\star \in W \}, \end{align*} $$
which we call a cut-and-project set. If W has nonempty interior,
${\Large \curlywedge }(W)$
, or any translate
$t+{\Large \curlywedge }(W)$
with
$t \in \mathbb {R}^{d}$
, is called a model set. When the boundary
$\partial W$
has zero Haar measure in H, the model set is called regular.
Now, one version of the Pisot substitution conjecture states that, if we have a set of control points
$\varLambda _{i}$
of a tiling arising from a Pisot substitution, we can find a suitable LCAG H and relatively compact sets
$W_{i} \subset H$
with nonempty interior such that
${\varLambda _{i} = {\Large \curlywedge }(W_{i})}$
up to a set of zero density. We restrict ourselves to unimodular substitutions, which means we have
$\lvert \det (M_{\varrho })\rvert = 1$
. Under this additional assumption, we obtain a Euclidean CPS [Reference Baake and Grimm13, Reference Sing36]. This is also plausible from the results in [Reference Barge and Kwapisz18], which establish an appropriate connection to a minimal rotation on a torus, and thus a fully Euclidean embedding.
Since one can, without loss of generality, assume that all
$\varLambda _{i}$
belong to the ring of integers
$\mathcal {O}_{\mathbb {Q}(\lambda )}$
of the algebraic number field
$\mathbb {Q}(\lambda )$
, where
$\lambda $
is the inflation factor, we employ a Minkowski embedding of
$\mathcal {O}_{\mathbb {Q}(\lambda )}$
to obtain the lattice
$\mathcal {L}$
. The projection
$\pi $
on its first coordinate gives the control points, whereas the remaining ones give the projection
$\pi _{\mathrm {int}}$
into the internal space, which is
$\mathbb {R}^{m}$
(if
$\lambda $
is of degree n, then
$m=n-1$
). Now, for the windows, one considers the
$\star $
-image of Equation (2-1) and its closure. Then, with
$W_{i} \mathrel {\mathop :}= \overline {\varLambda ^{\star }_i}$
,
$$ \begin{align} W_i \, = \bigcup_{j=1}^n \bigcup_{t \in T_{ij}} \lambda^{\star} W_j + t^{\star} , \end{align} $$
which is an iterated function system on the space
$(\mathcal {K}(\mathbb {R}^{m}))^n$
of all n-tuples of nonempty compact subsets of
$\mathbb {R}^{m}$
equipped with Hutchinson’s metric; see [Reference Baake, Moody, Baake and Moody15] for details. Since the RHS defines a contraction, by Hutchinson’s theorem [Reference Hutchinson24] (a version of Banach’s contraction principle), one has a unique solution. The resulting sets are called Rauzy fractals and have been studied extensively; see for example [Reference Arnoux and Harriss4, Reference Baake and Grimm14, Reference Siegel and Thuswaldner35]. Some of their properties include the fact that they are topologically regular (that is, no isolated points) and compact sets of positive measure with a (usually) fractal boundary [Reference Berthé, Siegel, Thuswaldner and Berthé19]. If the internal space is one-dimensional, one can even speak of the Rauzy fractal being a Cantorval, for which we refer the reader to [Reference Baake, Gorodetski and Mazáč10]. Let us summarise some useful properties of Rauzy fractals as follows.
Proposition 2.5 [Reference Sing36, Corollary 6.66].
Let
$\{W_{i} : 1 \leqslant i \leqslant n \}$
be the unique solution of Equation (2-2), as obtained from a unimodular Pisot substitution. Then, the following properties hold.
-
(i) All
$W_{i}$
have positive Lebesgue measure. -
(ii) In the IFS, Equation (2-2), the unions on the RHS are measure disjoint.
-
(iii) The boundaries
$\partial W_{i}$
have zero Lebesgue measure. -
(iv) The sets
$W_{i}$
are topologically regular, that is, they are the closures of their interiors and thus contain no isolated points. -
(v) The interiors of the
$W_{i}$
are disjoint.
In this context, we also refer the reader to further pioneering works on the geometry of unimodular Pisot substitutions, such as [Reference Arnoux and Ito5, Reference Ito and Rao25, Reference Sirvent and Wang37].
Since the topology of Rauzy fractals can be extremely complicated, such as having infinitely many connected components with noninteger boundary dimensions [Reference Siegel and Thuswaldner35], their covariograms are, by standard means, impossible to obtain. On the other hand, since Rauzy fractals correspond (in some sense) to fixed points of substitutions, we have an additional tool at hand: the pair-correlation functions. For a model set
$\varLambda = \bigcup _{j=1}^n \varLambda _{j}$
, let
$$ \begin{align*} \nu_{ij}(y) \, \mathrel{\mathop:}= \lim_{r\to \infty} \frac{1}{{\mathrm{card}} (\varLambda^{(r)})} \sum_{\substack{x \in \varLambda^{(r)}_{i}\\ x+y \in \varLambda_j}} \! \! 1 \end{align*} $$
denote the relative frequency of a two-point patch consisting of a control point of type i on the left and a control point of type j on the right with distance
$y \in \varLambda _j - \varLambda _{i}$
, where a negative y flips the role of left and right. This quantity exists uniformly by the uniform distribution property of model sets [Reference Baake and Grimm13, Theorem 7.3]. By applying the
$\star $
-map, with
$\varLambda ^{(r)}_j = \varLambda _{j} \cap B_{r}(0)$
, we obtain the representation
$$ \begin{align*} \nu_{ij}(y) \,& = \lim_{r\to \infty} \frac{1}{{\mathrm{card}} (\varLambda^{(r)})} \sum_{\substack{x^{\star} \in (\varLambda^{(r)}_{i})^{\star}\\ (x+y)^{\star} \in W_{j}}} \! \! 1 \; = \; \frac{1}{\text{vol}(W)} \int_{\mathbb{R}^{n-1}} \mathbf{1}_{W_{i}}(z) \, \mathbf{1}_{W_{j}}(z+y^{\star})\, \mathop{}\!{d} z \\ & = \, \frac{(\mathbf{1}_{-W_{i}} * \mathbf{1}_{W_{j}})(y^{\star})}{\text{vol}(W)} \, =\mathrel{\mathop:} \, g_{ij}(y^{\star}) , \end{align*} $$
where the second equality in the first line holds by Weyl’s theorem on uniform distribution [Reference Baake and Grimm13, Reference Kuipers and Niederreiter28]. Up to a factor of
${\mathrm {vol}}(W)^{-1}$
, the functions
$g_{ij}$
are precisely the covariograms of Definition 2.1, so
$$ \begin{align*} \nu_{ij}(x) = \begin{cases} \dfrac{1}{{\mathrm{vol}}(W)} \, (\boldsymbol{1}_{-W_{i}} \ast \boldsymbol{1}_{W_{j}})(x^{\star}) & \text{if } x \in \varLambda_{j}-\varLambda_{i}, \\ 0 & \text{otherwise}. \end{cases} \end{align*} $$
Remark 2.6. Recall that, by Proposition 2.2, the covariogram is a continuous function. Moreover, if
$x \in \varLambda _{j}-\varLambda _{i}$
, the points
$x^{\star }$
are dense in
$W_{j} - W_{i}$
, by definition of the CPS, and uniformly distributed by Weyl’s theorem [Reference Baake and Grimm13, Reference Kuipers and Niederreiter28]. Therefore, when plotting the covariogram on a sufficiently large sample of these
$x^{\star }$
, one expects a good illustration of the continuous function and its behaviour, though care is definitely required in view of the vast range of phenomena that are possible in the realm of continuous functions.
By realising the covariogram of the window as the autocorrelation function of the model set, we can circumvent issues with the window in internal space, and instead work with the renormalisation relations for
$\nu _{ij}$
. These techniques were introduced in [Reference Baake and Gähler7] and further developed and used, for example, in [Reference Baake, Gähler and Mañibo9, Reference Mañibo29, Reference Mazáč31]. We recall the main result that profits from the inflation structure and enables computing these quantities.
Proposition 2.7 ([Reference Mañibo29, Proposition 2.2.1] and [Reference Baake, Gähler and Mañibo9, Lemma 3.16]).
Let
$\{\varLambda _{i}\}_{i=1}^n$
be a fixed point of a primitive geometric inflation
$\varrho $
with inflation factor
$\lambda>1$
arising from a substitution over an n-letter alphabet. Then, the pair-correlation functions
$\nu _{ij}$
exist, and satisfy the exact renormalisation equations
$$ \begin{align*} \nu_{ij}(x) = \frac{1}{\lambda} \sum_{k, \ell =1}^n \, \sum_{r\in T_{ik}} \, \sum_{s\in T_{j\ell}} \nu_{k\ell} \,\bigg(\frac{x+r-s}{\lambda} \bigg) , \end{align*} $$
where
$T_{\varrho } = (T_{ij})$
is the displacement matrix of the inflation
$\varrho $
.
As
$\lambda>1$
, the arguments of the
$\nu _{ij}$
on the RHS of the equations in Proposition 2.7 are generally smaller in modulus than those on the left-hand side. Since
$\nu _{ij}(x) =0$
for all
$x\notin \varLambda _{j} - \varLambda _{i}$
, the system of equations splits naturally into a self-consistent part, which closes on itself, and the recursive part. As the correlation functions
$\nu _{ij}$
satisfy the renormalisation relations, their counterparts
$g_{ij}$
satisfy the same equations on a subset of
$\mathbb {R}^{m}$
given by the
$\star $
-image of
$\varLambda _{j} - \varLambda _{i}$
. These equations can also be derived without prior knowledge of the renormalisation equations from Proposition 2.7. It suffices to know the IFS for the windows, Equation (2-2), which gives some kind of a dual to the equations from Proposition 2.7. Indeed, the IFS can be turned into a set of equations for the characteristic functions of the windows. These, in turn, induce equations for their (cross) covarigrams, which we state next. We recall that
$\lambda ^{\star }$
refers to the linear mapping
$y \mapsto \lambda ^{\star }y$
in
$\mathbb {R}^m$
, which satisfies
$(\lambda x)^{\star } = \lambda ^{\star } x^{\star }$
for all
$x\in \mathbb {Q}(\lambda )$
.
Proposition 2.8. Let
$\varLambda _{i} = {\Large \curlywedge }(W_{i})$
be a collection of regular model sets arising from a single CPS
$(\mathbb {R}, \, \mathbb {R}^{m}, \, \mathcal {L})$
describing the control points of a fixed point of a unimodular inflation with a PV inflation factor
$\lambda $
, and with displacement matrix
$T_{\varrho } = (T_{ij})$
. Then, the functions
$g_{\alpha \beta } \mathrel {\mathop :}= \boldsymbol {1}_{-W_{\alpha }} \ast \boldsymbol {1}_{W_{\beta }}$
satisfy, for almost all
$y \in \mathbb {R}^{m}$
, the relations
$$ \begin{align*} g_{ij}(y) = \frac{1}{\lambda} \sum_{k, \ell=1}^{n} \, \sum_{r\in T_{ik}} \, \sum_{s \in T_{j\ell}} \, g_{k \ell} ( ({\lambda^{\star}})^{-1}(y+ r^{\star} {-} s^{\star})). \end{align*} $$
Proof. We recall the properties of the covariogram function from Proposition 2.2 and observe that
$(\boldsymbol {1}_{-\lambda ^{\star } A} \, \ast \, \boldsymbol {1}_{\lambda ^{\star }B})(y) \, = |\!\det (\lambda ^{\star }) | ( \boldsymbol {1}_{-A} \ast \boldsymbol {1}_{B} ) (({\lambda ^{\star }})^{-1}y)$
holds for arbitrary compact sets
$A,B$
, which follows by a standard change of the variable argument. We also recall that the unions on the RHS of Equation (2-2) are measure disjoint and their boundary is of Lebesgue measure zero (Proposition 2.5). Then,
$$ \begin{align*} g_{ij}(y)\, & = \, (\boldsymbol{1}_{-W_{i}} \, \ast \, \boldsymbol{1}_{W_j}) (y) = (\boldsymbol{1}_{-\bigcup_{k=1}^n \bigcup_{r \in T_{ik}} \lambda^{\star} W_k + r^{\star}} \, \ast \, \boldsymbol{1}_{\bigcup_{\ell=1}^n \bigcup_{s \in T_{j\ell}} \lambda^{\star} W_{\ell} + s^{\star}} ) (y)\\ & = \sum_{k, \ell=1}^{n} \, \sum_{r\in T_{ik}} \, \sum_{s \in T_{j\ell}} \, \lvert \det (\lambda^{\star}) \rvert \, (\boldsymbol{1}_{-W_{k}} \ast \,\boldsymbol{1}_{W_{\ell}}) ( ({\lambda^{\star}} )^{-1} (y+ r^{\star} {-} s^{\star}) ) \\ & = \, \frac{1}{\lambda} \sum_{k,\, \ell=1}^{n} \, \sum_{r\in T_{ik}} \, \sum_{s \in T_{j\ell}} \, g_{k\ell}( ({\lambda^{\star}})^{-1} (y+ r^{\star} {-} s^{\star}) ), \end{align*} $$
which holds almost everywhere. For the last equality, let us recall that the Pisot property and unimodularity together imply that
$\lambda ^{-1} = \lvert \det (\lambda ^{\star }) \rvert $
.
We refer the reader to [Reference Mazáč30] for a brief summary of the nonunimodular setting with LCAG H having a p-adic component. Let us turn to concrete examples and explain further details.
3 Binary alphabet examples
In this section, we provide some examples of determining the covariograms of Rauzy fractal windows from Pisot substitutions over a binary alphabet. We first construct a model set with a highly irregular window, and implement the renormalisation procedure to calculate the covariogram. Note that in this one-dimensional internal space case, as conjectured in [Reference Baake, Gorodetski and Mazáč10, Reference Mazáč30], there are two possibilities for the covariogram: a piecewise linear function, arising from interval-type windows, or a ‘spiky’ function, as illustrated in Figure 3. The latter arises when the window is a Cantorval. For an example where the internal space is two-dimensional, we refer the reader to [Reference Klick26].
3.1 The sister silver mean tiling
Let us construct the sister silver mean (SSM) chain. Consider the binary alphabet
$\mathcal {A} = \{a,b\}$
, and the substitution rule
$$ \begin{align*} \varrho_{s} \,:\, \begin{cases} a \mapsto bba ,\\ b \mapsto ab , \end{cases} \end{align*} $$
which we abbreviate as
$\varrho _{s} = (bba,ab) = (\varrho _{s}(a), \varrho _{s}(b))$
. Starting with the legal seed
$\omega _{(0)} = a|a$
, where
$|$
denotes the origin, by iteration one obtains a sequence of growing words
$\omega _{(n)}$
which converges, with obvious meaning in the product topology, towards a two cycle of bi-infinite words
$\omega , \omega ^{\prime } \in \mathcal {A}^{\mathbb {Z}}$
. The left-infinite half is fixed, while the right-infinite half alternates between two words that differ on a set of zero density only. In fact, they form a proximal pair; see [Reference Baake and Grimm13, Section 4.3] for details.
Now, we can move to the geometric description and easily deal with this difference in the cut-and-project description. The substitution matrix of
$\varrho _{s}$
is
$ M_{s} =\big (\begin {smallmatrix}1 & 1\\ 2 & 1\end {smallmatrix}\big )$
, with PF eigenvalue
$\lambda = 1+\sqrt {2}$
. The left and (statistically normalised) right PF eigenvectors are
$$ \begin{align} v_{\ell} = (\!\sqrt{2},\, 1) \quad \text{ and } \quad v_{r} = (\lambda^{-1}, \,\lambda^{-1}\sqrt{2}\, )^{\top} =\,(\lambda-2, \,3-\lambda\, )^{\top}. \end{align} $$
The a and b prototiles for this choice have lengths
$\sqrt {2}$
and
$1$
. The inflated a-tile is called the (level-one) a-supertile, and, analogously, we have the (level-one) b-supertile. Below, they will be called the a and b supertiles, and denoted by
$\overline {a}$
and
$\overline {b}$
, respectively; the inflation rule is depicted in Figure 1. Denote the set of left endpoints (control points) of tilings corresponding to
$\omega $
and
$\omega ^{\prime }_{}$
by
$\varLambda _{\omega }$
and
$\varLambda _{\omega ^{\prime }_{}}$
. These sets are in a natural one-to-one correspondence with
$\omega $
and
$\omega ^{\prime }_{}$
, and each has a density of
$({2 + \sqrt {2} })/{4}$
. Thus, the discrepancies between the two fixed points form a zero density set; compare [Reference Baake and Grimm13, Ch. 4].
Geometric inflation rule for the self-similar SSM tiling.
Denote the sets of control points of the tiles of types a and b by
$ \varLambda _{a}$
and
$ \varLambda _{b}$
, with
$ {\varLambda = \varLambda _{a} \,\dot {\cup }\, \varLambda _{b}}$
. By our choice of the control point, it immediately follows that
$\varLambda \subset \mathbb {Z}[\sqrt {2} \,]$
, which is the smallest
$\mathbb {Z}$
-module into which we can embed
$\varLambda $
. The standard Minkowski embedding of
$\varLambda \subset \mathbb {Z}[\sqrt {2} \,]$
gives the lattice
$\mathcal {L} = \mathbb {Z}(1,1) \oplus \mathbb {Z}(\sqrt {2}, -\sqrt {2})$
; see [Reference Baake and Grimm13, Ch. 7] and [Reference Koch27, Theorem. 2.13.1] for details. In particular, the
$\star $
-map is given by the nontrivial Galois automorphism of
$\mathbb {Q}(\sqrt {2})$
, which is
$a+b\sqrt {2} \mapsto (a+b\sqrt {2})^{\star } = a-b\sqrt {2}$
. This gives the Euclidean CPS
By construction,
$\pi _{\text {int}}(\mathcal {L})$
is dense in internal space. From a CPS of the form of Equation (3-2), we can construct a regular model set, with suitable
$W \subset \mathbb {R}$
, as
$$ \begin{align*} {\Large \curlywedge}(W)\, \mathrel{\mathop:}= \, \{x \in L : x^{\star} \in W\} \, \subset \, \mathbb{Z}[\!\sqrt{2}\,]. \end{align*} $$
As described in Section 2, the window corresponding to the fixed point of the inflation rule, up to the zero-density set of discrepancies, is found by first considering the following set-valued iterations,
$$ \begin{align*} \begin{split} \varLambda_{a} \, & =\, ( \lambda \varLambda_{a} + 2 ) \; \dot{\cup}\; \lambda \varLambda_{b} , \\ \varLambda_{b} \, & =\, \lambda \varLambda_{a} \; \dot{\cup} \; ( \lambda \varLambda_{a} + 1 ) \; \dot{\cup} \; ( \lambda \varLambda_{b} + \lambda - 1). \end{split} \end{align*} $$
The RHSs describe the positions of the tiles in the inflated version of the tilings. For example, applying the inflation rule to an a tile produces another a preceded by two b’s, giving the first term in the first equation. Similarly, inflating a b tile produces an a at the inflated point, giving the last term in the first equation. Taking the
$\star $
-image and closure of the sets, denoted
$W_{a,b}:= \overline { \varLambda ^{\star }_{a,b}}$
, gives the following internal space IFS:
$$ \begin{align} \begin{aligned} W_{a} \, & =\, ( (2-\lambda) W_{a} + 2 )\, \cup\, (2-\lambda) W_{b} , \\ W_{b} \, & = \, (2-\lambda) W_{a} \, \cup\, ( (2-\lambda) W_{a} + 1 ) \, \cup\, ( (2-\lambda) W_{b} + 1 - \lambda). \end{aligned} \end{align} $$
The fixed point of this contractive IFS, as
$\lambda ^{\star }<1$
, is a Rauzy fractal; see Figure 2. By Proposition 2.5, these windows define regular model sets, and so does their union.
The windows of the SSM inflation tiling, which are the fixed point of the IFS, Equation (3-3);
$W_{a}$
is red (top) and
$W_{b}$
is blue (bottom). The windows are one-dimensional, but we assign some fixed arbitrary height to the points for illustration. The windows are measure-theoretically disjoint, but the resolution is limited by the large Hausdorff dimension of the window boundaries.
There are several ways to determine the Hausdorff dimension of the boundary of the window. One method, laborious but allowing a complete reconstruction of the window, involves the graph iterated function system of the window; see [Reference Baake, Gorodetski and Mazáč10, Reference Sing36]. Here, we use the overlap algorithm from [Reference Akiyama and Lee2, Reference Baake, Gähler and Gohlke8, Reference Solomyak38], based on the discrepancy matrix; this is applicable to more complex tilings, but we need only consider our one-dimensional case. Consider two overlaid copies of the SSM tiling, one being a copy shifted by an element of the return module. We say a pair of tiles, one from each copy, form an overlap if the intersection of their supports has a nonempty interior. We can then consider the overlap itself as a tile determined by the support, the type of tiles from which it comes, and the offset of the two tiles. If the pair of tiles have the same type and position, we call it a coincidence overlap; otherwise, it is referred to as a discrepancy overlap. Therefore, from the original substitution, we can define another substitution on the overlaps; the corresponding substitution matrix is called the discrepancy matrix.
The Hausdorff dimension of the boundary
$\partial W$
is then given [Reference Baake, Gähler and Gohlke8, Proposition 6.1] by
$$ \begin{align*} \dim_{\mathrm{H}} (\partial W) = \overline{\dim}_{\mathrm{H}} (\partial W) \, =\, \frac{ \, \log (\gamma)}{\log (\lambda_{\mathrm{PF}})} , \end{align*} $$
where
$\overline {\dim }_{\mathrm {H}} (\partial W)$
is the maximum of
$\overline {\dim }_{\mathrm {H}} (\partial W_{a,b})$
,
$\lambda _{\mathrm {PF}}$
is the PF eigenvalue of the substitution matrix, and
$\gamma $
is the spectral radius of the discrepancy matrix. The first equality holds because
$\varLambda _{a}$
and
$\varLambda _{b}$
are MLD to
$\varLambda $
, see [Reference Baake, Gorodetski and Mazáč10] for details, thus the Hausdorff dimension of
$\partial W_{a}$
,
$\partial W_{b}$
and
$\partial W$
must be equal [Reference Baake and Grimm13, Remark 7.6]. Implementing this algorithm, an elementary calculation leads to the discrepancy matrix
$$ \begin{align*} M_{\text{dc}} \,= \, \begin{pmatrix} 1& 0&0&0&1&1&0\\ 1& 0&0&0&0&0&0\\ 1& 0&0&0&0&0&0\\ 1& 0&0&0&0&0&0\\ 0& 1&0&0&0&0&0\\ 0& 0&1&0&0&1&1\\ 0& 0&1&0&1&1&0 \end{pmatrix} \end{align*} $$
with characteristic polynomial 
. The spectral radius of
$M_{\text {dc}}$
thus has a value of approximately
$2.205\,57$
, giving
$$ \begin{align*} \dim_{\mathrm{H}} (\partial W) \,=\, \frac{\log (\gamma)}{\log (\lambda_{\mathrm{PF}})} \, \approx \, 0.897\,45. \end{align*} $$
Obviously, computing the covariogram of this window according to Definition 2.1 looks difficult if not impossible. To overcome this problem, we now introduce the renormalisation procedure. For the pair correlations
$\nu _{\alpha \beta }$
, where
$\alpha , \beta \in \{a,b\}$
, we have
$\nu _{\alpha \beta }(z) = 0$
for any
$z \notin \varLambda _{\beta }- \varLambda _{\alpha }$
and
$\nu _{\alpha \beta }(0)=0$
for
$\alpha \neq \beta $
. Moreover, the right PF eigenvector of Equation (3-1) gives
$\nu _{aa}(0)=\lambda ^{-1} = \lambda -2$
and
$\nu _{bb}(0)=\lambda ^{-1}\hspace{-0.5pt}\!\sqrt {2}=3-\lambda $
. We also have the symmetry relation
$\nu _{\alpha \beta }(-z)= \nu _{ \beta \alpha }(z)$
, and the summatory relationship
$$ \begin{align*} \nu(z) \,=\, \nu_{aa}(z) \,+\, \nu_{ab}(z)\,+\, \nu_{ba}(z)\,+\, \nu_{bb}(z) \end{align*} $$
from Proposition 2.2(vi).
With the symmetry relations and inflation structure, we can determine the renormalisation relations of
$\nu _{\alpha \beta }$
as a special case of Proposition 2.7; see [Reference Klick26] for worked examples.
Theorem 3.1. The pair correlations
$\nu _{\alpha \beta }$
with
$\alpha , \beta \in \{a,b\}$
of the SSM tiling satisfy the exact renormalisation relations
$$ \begin{align*} \lambda\nu_{aa}(z)\, &= \,\nu_{aa}\,\bigg(\frac{z}{\lambda}\bigg) \,+\, \nu_{ab}\,\bigg(\frac{z+2}{\lambda}\bigg) \,+\, \nu_{ba}\,\bigg(\frac{z-2}{\lambda}\bigg) \,+\, \nu_{bb}\,\bigg(\frac{z}{\lambda}\bigg) ,\\[2mm] \lambda\nu_{ab}(z)\, &= \, \nu_{aa}\,\bigg(\frac{z+2}{\lambda}\bigg)\,+\, \nu_{aa}\,\bigg(\frac{z+1}{\lambda}\bigg)\,+\, \nu_{ab} \,\bigg(\frac{z+3-\lambda}{\lambda}\bigg)\,+\, \nu_{ba}\,\bigg(\frac{z}{\lambda}\bigg)\\ &\phantom{==}\,+\, \nu_{ba}\,\bigg(\frac{z-1}{\lambda}\bigg)\,+\, \nu_{bb} \,\bigg(\frac{z+1-\lambda}{\lambda}\bigg) ,\\[2mm] \lambda\nu_{ba}(z)\, &= \, \nu_{aa}\,\bigg(\frac{z-2}{\lambda}\bigg)\,+\, \nu_{aa}\,\bigg(\frac{z-1}{\lambda}\bigg)\,+\, \nu_{ab}\,\bigg(\frac{z+1}{\lambda}\bigg) \,+\, \nu_{ab}\,\bigg(\frac{z}{\lambda}\bigg)\\ &\phantom{==}\,+\, \nu_{ba}\,\bigg(\frac{z+\lambda-3}{\lambda}\bigg)\,+\, \nu_{bb}\,\bigg(\frac{z-1+\lambda}{\lambda}\bigg) , \\[2mm] \lambda\nu_{bb}(z)\, &= \,2 \nu_{aa} \,\bigg(\frac{z}{\lambda}\bigg)\,+\, \nu_{aa}\,\bigg(\frac{z+1}{\lambda}\bigg)\,+\, \nu_{aa}\,\bigg(\frac{z-1}{\lambda}\bigg)\,+\, \nu_{ab}\,\bigg(\frac{z+1-\lambda}{\lambda}\bigg) \\ &\phantom{==}\,+\, \nu_{ab}\,\bigg(\frac{z+2-\lambda}{\lambda}\bigg)\,+\, \nu_{ba}\,\bigg(\frac{z+\lambda-2} {\lambda}\bigg)\,+\, \nu_{ba} \,\bigg(\frac{z-1+\lambda} {\lambda}\bigg)\,+\, \nu_{bb}\,\bigg(\frac{z+\lambda}{\lambda} \bigg) , \end{align*} $$
with
$z \in \mathbb {Z}[\sqrt {2}\,]$
,
$\lambda = 1+\sqrt {2}$
, and
$\nu _{\alpha \beta }(z) =0$
for
$z \notin \varLambda _{\beta }- \varLambda _{\alpha }$
.
This is an infinite set of linear equations. However, via the inflation structure, all arguments with
$|z|> \lambda +1$
are recursively determined from the self-consistent part of the equations. As this is used later, we include the derivation. For a similar example, where the inflation factor is non-Pisot, we refer the reader to [Reference Baake, Frank, Grimm and Robinson6]. We remark that renormalisation does not need local recognisability, see [Reference Baake, Gähler and Mañibo9], but it makes the process easier.
Proposition 3.2. Consider the renormalisation relations of Theorem 3.1 with the symmetry and vanishing conditions, and the arguments restricted to
$z \in \varLambda - \varLambda $
with
$|z| \leqslant \lambda +1$
. This is a finite, closed set of equations with a one-dimensional solution space. In particular, once
$\nu _{aa}(0)$
is given, the solution is unique.
Proof. By symmetry, we need only to look at
$z\geqslant 0$
, and when
$z\leqslant \lambda +1$
, no argument on the RHS of the identities in Theorem 3.1 exceeds
$\lambda +1$
. We thus consider
$$ \begin{align*} \varLambda- \varLambda \cap [0, \lambda+1] = \{0,\, 1,\, 2 ,\,3 ,\, \lambda-1,\, \lambda ,\, 2\lambda -2,\, \lambda+1 \} , \end{align*} $$
which gives the first claim. We omit any values prohibited by the tile geometry, as all pair correlations would then be zero. The values for
$z=0$
follow from the PF eigenvector, and the fact that a tile can only have one type. The geometry of the tiles implies the other occurrences of
$0$
in Table 1; however, these can also be checked using the relations from Theorem 3.1 and the vanishing conditions. Note that
$\lambda ^{-1} = \lambda -2$
,
$\lambda ^{2} = 1+2\lambda $
and
$\lambda ^{\star } = - \lambda ^{-1}$
. Using these relations, we can easily solve the resulting finite set of equations by hand, the details of which are omitted. In particular,
$\nu _{aa}(0)$
is not fixed by the relations, and all other values can be written in terms of
$\nu _{aa}(0)$
; thus, the solution space is indeed one-dimensional.
The self-consistent part of the renormalisation equations for the SSM inflation tiling. Impossible distances (whose pair correlations evaluate to
$0$
due to the tile geometry) are omitted. The column with
$z=0$
contains the relative tile frequencies, as given by the right PF eigenvector of
$M_{\varrho }$
.
Uniqueness follows from
$\nu _{aa}(0)+\nu _{bb}(0)=1$
, with the values listed in Table 1, where the interpretation of
$\nu _{aa}(0)$
and
$\nu _{bb}(0)$
as the relative tile frequencies is used.
A similar proof was given in [Reference Klick26], to which we refer the reader for further details and examples. Via the correspondence between the equations from Propositions 2.7 and 2.8, this also gives us the renormalisation approach to the covariogram functions
$g_{\alpha \beta }$
. We can now plot the covariograms using the recursive structure of the equations, with the self-consistent part serving as initial values. The result is shown in Figure 3. Despite their appearance, by Proposition 2.2, each function is continuous and also bounded above by a tent function. Moreover, as mentioned in Remark 2.6, Figure 3 is an acceptable illustration of the covariogram as we know its exact value at a dense and uniformly distributed set of points, in combination with the continuity of the function.
Representation of the window covariograms of the SSM system. The red function is
$g_{aa}$
,
$g_{bb}$
is blue,
$g_{ab}$
is black,
$g_{ba}$
is grey, and the covariogram g of the total window is pink (top). The plot of g consists of
$35\,323$
points; we also include a closer view of a small section to illustrate the irregular behaviour.
3.2 A more complex example
Next, we construct another window system, with the same CPS, whose boundaries exhibit even stronger fractal behaviour than that of the SSM system. Consider the substitution
$$ \begin{align*} \sigma \,:\, \begin{cases} a \mapsto aaaaabb ,\\ b \mapsto baa , \end{cases} \end{align*} $$
with corresponding substitution matrix
$M_{\sigma } =\big (\begin {smallmatrix} 5 & 2\\ 2 & 1\end {smallmatrix}\big )$
and PF eigenvalue
$$ \begin{align*} \lambda_{\mathrm{PF}} = \lambda^{2} = 2\lambda +1 =3+2\sqrt{2}. \end{align*} $$
The left and (statistically normalised) right PF eigenvectors are given by
$$ \begin{align*} w_{\ell} = ( \lambda, 1) \quad \text{and} \quad w_{r} = \bigg(\frac{\lambda-1}{2}, \,\frac{3-\lambda}{2} \bigg)^{\top}. \end{align*} $$
This substitution produces a tiling of
$\mathbb {R}$
in an analogous way as before. To determine the window, we go through the same procedure as in the previous section. The relations for the sets
$ \varLambda _{a}$
and
$ \varLambda _{b}$
are given by
$$ \begin{align*} \begin{split} \varLambda_{a} \, & =\,\lambda^{2} \varLambda_{a} \; \dot{\cup}\; ( \lambda^{2} \varLambda_{a} \,{{+}}\, \lambda ) \; \dot{\cup}\; (\lambda^{2} \varLambda_{a} \,{{+}}\, 2 \lambda ) \,\,\dot{\cup}\, (\lambda^{2} \varLambda_{a} \,{{+}}\, 3 \lambda ) \\[3pt] & \quad\quad {\dot{\cup}}\, ( \lambda^{2} \varLambda_{a} \,{{+}}\, 4 \lambda ) \; \dot{\cup}\; ( \lambda^{2} \varLambda_{b} \,{{+}}\, 1 ) \; \dot{\cup} \; ( \lambda^{2} \varLambda_{b} \,{{+}}\, \lambda \,{{+}}\,1 ) , \\[5pt] \varLambda_{b} \, & =\, (\lambda^{2} \varLambda_{a} \,{{+}}\, 5 \lambda ) \; \dot{\cup}\; ( \lambda^{2} \varLambda_{a} \,{{+}}\, 5 \lambda \,{{+}}\,1 ) \; \dot{\cup}\; ( \lambda^{2} \varLambda_{b} ). \end{split} \end{align*} $$
Again, taking
$W_{a,b}:= \overline { \varLambda ^{\star }_{a,b}}$
gives the following internal space IFS:
$$ \begin{align*} \begin{split} W_{a} \, & =\,\lambda^{-2} W_{a} \,\cup\, ( \lambda^{-2} W_{a} \,{{-}}\, \lambda^{-1} ) \,\cup\, (\lambda^{-2} W_{a} \,{{-}}\, 2 \lambda^{-1} ) \,\cup\, (\lambda^{-2} W_{a} \,{{-}}\, 3 \lambda^{-1} ) \\[3pt] & \quad \quad \;\;\, \cup\, ( \lambda^{-2} W_{a} \,{{-}}\, 4\, \lambda^{-1} ) \,\cup\, ( \lambda^{-2} W_{b} \,{{+}}\, 1 ) \,\,\cup\, ( \lambda^{-2} W_{b} \,{{+}}\, 1\,{{-}}\,\lambda^{-1} ) ,\\[5pt] W_{b} \, & =\, (\lambda^{-2} W_{a} \,{{-}}\, 5 \lambda^{-1} ) \,\cup\, ( \lambda^{-2} W_{a} \,{{+}}\, 1\,{{-}}\, 5 \lambda^{-1} ) \,\cup\, ( \lambda^{-2} W_{b} ). \end{split} \end{align*} $$
The fixed point of this contractive IFS is again the window system, illustrated in Figure 4. Via the overlap algorithm, the dimension of the fractal boundary is determined to be
$$ \begin{align*} \dim_{\mathrm{H}} (\partial W) \,=\, \frac{ \, \log (\gamma)}{\log (\lambda_{\mathrm{PF}})} \, \approx \, 0.966\,473\,9, \end{align*} $$
where
$\gamma \approx 5.493\,959$
is the largest root of
$x^{3}-7x^{2}+7x+7$
, the characteristic polynomial of the discrepancy matrix. The renormalisation relations are as follows.
Illustration of the windows for the substitution
$\sigma $
;
$W_{a}$
is red (top) and
$W_{b}$
is blue (bottom). As before, the windows are one-dimensional, but we assign some fixed arbitrary height to the points for illustration. Furthermore, by Proposition 2.5, the windows are measure-theoretically disjoint, but we are limited by the resolution of the plot and the large Hausdorff dimension of the window boundaries.
Theorem 3.3. The pair correlations
$\nu _{\alpha \beta }$
with
$\alpha , \beta \in \{a,b\}$
of the tiling induced by
$\sigma $
satisfy the exact renormalisation relations
$$ \begin{align*} \lambda^{2} \nu_{aa}(z)\, &= \, 5 \nu_{aa}\,\bigg(\frac{z}{\lambda^{2}}\bigg) + 4 \nu_{aa}\,\bigg(\frac{z-\lambda}{\lambda^{2}}\bigg) + 4 \nu_{aa}\,\bigg(\frac{z+\lambda}{\lambda^{2}}\bigg) + 3 \nu_{aa}\,\bigg(\frac{z-2\,\lambda}{\lambda^{2}}\bigg) + \nu_{bb}\,\bigg(\frac{z+\lambda}{\lambda^{2}}\bigg)\\ &\phantom{==} + 3 \nu_{aa} \,\bigg(\frac{z+2\lambda}{\lambda^{2}}\bigg) + 2 \nu_{aa}\,\bigg(\frac{z-3\,\lambda}{\lambda^{2}}\bigg) + 2 \nu_{aa}\,\bigg(\frac{z+3\,\lambda}{\lambda^{2}}\bigg) + \nu_{aa}\,\bigg(\frac{z-4\,\lambda}{\lambda^{2}}\bigg)\\ &\phantom{==} + \nu_{aa} \,\bigg(\frac{z+4\,\lambda}{\lambda^{2}}\bigg) + 2 \nu_{ab}\,\bigg(\frac{z-1}{\lambda^{2}}\bigg) + 2 \nu_{ba}\,\bigg(\frac{z+1}{\lambda^{2}}\bigg) + \nu_{ab}\,\bigg(\frac{z-\lambda-1}{\lambda^{2}}\bigg)\\ &\phantom{==} + \nu_{ba}\,\bigg(\frac{z+\lambda+1}{\lambda^{2}}\bigg) + \nu_{ab}\,\bigg(\frac{z+4\,\lambda-1}{\lambda^{2}}\bigg) + \nu_{ba}\,\bigg(\frac{z-4\,\lambda+1}{\lambda^{2}}\bigg) + \nu_{bb} \,\bigg(\frac{z-\lambda}{\lambda^{2}}\bigg) \\ &\phantom{==} + 2 \nu_{ab}\,\bigg(\frac{z+\lambda-1}{\lambda^{2}}\bigg) + 2\nu_{ba}\,\bigg(\frac{z-\lambda+1}{\lambda^{2}}\bigg) + 2\nu_{ab} \,\bigg(\frac{z+2\lambda-1}{\lambda^{2}}\bigg) + 2 \nu_{bb} \,\bigg(\frac{z}{\lambda^{2}}\bigg)\\ &\phantom{==} + 2 \nu_{ba} \,\bigg(\frac{z-2\,\lambda+1}{\lambda^{2}}\bigg) + 2\nu_{ab}\,\bigg(\frac{z+3\,\lambda-1}{\lambda^{2}}\bigg) + 2\nu_{ba}\,\bigg(\frac{z-3\,\lambda+1}{\lambda^{2}}\bigg),\\[2mm] \lambda^{2} \nu_{bb}(z)\, &= \,2 \nu_{aa} \,\bigg(\frac{z}{\lambda^{2}}\bigg) + \nu_{aa}\,\bigg(\frac{z-1}{\lambda^{2}}\bigg) + \nu_{ab}\,\bigg(\frac{z+5\, \lambda} {\lambda^{2}}\bigg) + \nu_{aa}\,\bigg(\frac{z+1}{\lambda^{2}}\bigg) \\ &\phantom{==} + \nu_{ab}\,\bigg( \frac{z+5\, \lambda^{}+1}{\lambda^{2}}\bigg) + \nu_{ba}\,\bigg(\frac{z-5\,\lambda^{}} {\lambda^{2}}\bigg) + \nu_{ba}\,\bigg(\frac{z-5\,\lambda^{}-1} {\lambda^{2}}\bigg) + \nu_{bb}\,\bigg(\frac{z}{\lambda^{2}}\bigg), \\[2mm] \lambda^{2}\nu_{ab}(z)\, &= \,\nu_{aa}\,\bigg(\frac{z-5\lambda}{\lambda^{2}}\bigg) + \nu_{aa}\,\bigg(\frac{z-5\lambda-1}{\lambda^{2}}\bigg) + \nu_{ab}\,\bigg(\frac{z}{\lambda^{2}}\bigg) + \nu_{aa} \,\bigg(\frac{z-4\lambda}{\lambda^{2}}\bigg) + \nu_{ba} \,\bigg(\frac{z-4\lambda} {\lambda^{2}}\bigg)\\ &\phantom{==} + \nu_{aa} \,\bigg(\frac{z-4\lambda-1} {\lambda^{2}} \bigg) + \nu_{ab}\,\bigg(\frac{z+\lambda}{\lambda^{2}}\bigg) + \nu_{aa} \,\bigg(\frac{z-3\lambda}{\lambda^{2}}\bigg) + \nu_{aa}\,\bigg(\frac{z-3\lambda-1} {\lambda^{2}}\bigg) \\ &\phantom{==} + \nu_{ab} \,\bigg(\frac{z+2\lambda}{\lambda^{2}}\bigg) + \nu_{aa}\,\bigg(\frac{z-2\lambda}{\lambda^{2}}\bigg) + \nu_{aa} \,\bigg(\frac{z-2\lambda-1}{\lambda^{2}}\bigg) + \nu_{ab}\,\bigg(\frac{z+3\lambda}{\lambda^{2}}\bigg) \\ &\phantom{==} + \nu_{aa} \,\bigg(\frac{z-\lambda}{\lambda^{2}}\bigg) + \nu_{aa}\,\bigg(\frac{z-\lambda-1} {\lambda^{2}}\bigg) + \nu_{ab}\,\bigg(\frac{z+4\lambda} {\lambda^{2}}\bigg) + \nu_{ba} \,\bigg(\frac{z+1-5\lambda}{\lambda^{2}}\bigg)\\ &\phantom{==} + \nu_{ba} \,\bigg(\frac{z-5\lambda} {\lambda^{2}}\bigg) + \nu_{bb}\,\bigg(\frac{z+1} {\lambda^{2}}\bigg) + \nu_{ba} \,\bigg(\frac{z+1-4\lambda}{\lambda^{2}}\bigg) + \nu_{bb}\,\bigg(\frac{z+\lambda+1} {\lambda^{2}}\bigg) , \end{align*} $$
together with
$\nu _{ba}(z) = \nu _{ab}(-z)$
,
$z \in \mathbb {Z}[\sqrt {2}\,]$
,
$\lambda = 1+ \sqrt {2}$
, and
$\nu _{\alpha \beta }(z) =0$
for
$z \notin \varLambda _{\beta }- \varLambda _{\alpha }$
.
The self-consistent part of the renormalisation equations for the tiling arising from
$\sigma $
, with the natural tile lengths given by the left PF eigenvector. Distances that are not possible, that is, ones for which all pair correlations evaluate to zero due to the tile geometry, are omitted. The column corresponding to
$z=0$
contains the relative tile frequencies, as given by the (statistically normalised) right PF eigenvector of
$M_{\sigma }$
.
The self-consistent part of the relations is listed in Table 2. As the calculations are analogous to Proposition 3.2, we omit a proof.
Corollary 3.4. Consider the renormalisation relations of Theorem 3.3 with the symmetry and vanishing conditions, and the arguments restricted to
$z \in \varLambda - \varLambda $
with
$|z| \leqslant 2\,\lambda +1$
. This is a finite, closed set of equations with a one-dimensional solution space. In particular, once
$\nu _{aa}(0)$
is given, the solution is unique.
The covariogram can now be computed as before, and is shown in Figures 5 and 6. We remark again that this is indeed a continuous function, despite its highly discontinuous appearance. To our knowledge, this is the first example of such a covariogram having a ‘splitting’ behaviour. One can do the following as a consistency check for the continuity. Take a large patch of the tiling, approximately
$500\,000$
tiles, and construct the set of tile coordinates. The Minkowski difference of this set is then taken with itself; with the above number of tiles, it should produce approximately five million distinct distances. To efficiently run this set of distances through the relations from Theorem 3.3, one can implement a ‘valid distance’ test; this checks whether the integer coordinates of
$z= a \lambda + b$
have the same parity (or are zero) and whether
$z^{\star }$
lies in a bounded region of internal space. The distance z is then passed to the relations only if both requirements are met. Additionally, running the relations in parallel significantly improves computation time.
Point plot, with
$188\,214$
points, of the covariogram of the window of Figure 4. Here, the splitting behaviour is highlighted; the distances involving an even number of b’s are blue (top), and those with an odd number are red (bottom).
Point plot, derived from
$4\,826\,361$
points, of the covariogram of the window of Figure 4. The closer view illustrates the slow convergence and complex behaviour of the function.
With these large samples, one sees that the gap will close, albeit incredibly slowly; Figure 6 illustrates this behaviour. We emphasise again, as in Remark 2.6 and in the previous example, that this is a valid approximation of the continuous function as we are sampling from a dense and uniformly distributed subset of the support. The upper and lower branches of the split can be identified as those distances that contain, respectively, an even and odd number of b’s. This behaviour is thus a consequence of the combinatorics of the substitution and not triggered by the high boundary dimension of the window, as one might first guess. This is well worth further examination, but is far beyond the scope of this brief note. Also, with regard to the original question of a natural counterpart to the theta series of periodic point sets, the complex behaviour of the covariograms will also call for a renewed, and perhaps conceptually different, approach to aperiodic theta series.
Our above analysis was based on one-dimensional inflation tilings with an internal space that is also one-dimensional. It is an obvious task to extend both to higher dimensions. In a first step, one should consider unimodular inflation tilings with a ternary alphabet, thus giving Rauzy fractals in
$\mathbb {R}^2$
. First results on the Kolakoski sequence are presented in [Reference Klick26], and the same type of analysis can be done for the classic Tribonacci sequence. Computationally, both are significantly more involved yet of limited value because the windows are simply connected sets with a fractal boundary. The resulting covariograms still look well behaved and mainly smooth. More interesting would be the twisted Tribonacci sequence [Reference Arnoux and Ito5, Reference Baake, Gähler and Gohlke8], which leads to a disconnected window system that may be considered as an analogue of a Cantorval. More work is needed to explore the possibilities here.
Acknowledgements
It is our pleasure to thank Claudia Alfes and Paul Kiefer for helpful discussions, and Michael Coons and Nicolae Strungaru for useful hints on the manuscript. We are grateful to an anonymous referee for several careful and constructive comments, which helped us to improve the presentation. M.B. is grateful to the School of Mathematics and Physics of the University of Tasmania in Hobart, where this manuscript was completed, for their hospitality.
Open access
