MATHEMATICAL DIFFRACTION THEORY OF
DETERMINISTIC AND STOCHASTIC STRUCTURES:
AN INFORMAL SUMMARYMICHAEL BAAKE AND UWE GRIMM

Abstract. Mathematical diffraction theory is concerned with the determinationof the diffraction image of a given structure and the corresponding inverse problem
of structure determination. In recent years, the understanding of systems with continuous and mixed spectra has improved considerably. Moreover, the phenomenonof homometry shows various unexpected new facets. This is particularly so whensystems with stochastic components are taken into account.
After a brief introduction and a summary of pure point spectra, we discussclassic deterministic examples with singular or absolutely continuous spectra. In
particular, we present an isospectral family of structures with continuously varying
entropy. We augment this with more recent results on the diffraction of dynamicalsystems of algebraic origin and various further systems of stochastic nature. A
systematic approach is mentioned via the theory of stochastic processes.

Contents
1. Introduction to diffraction theory
2. Pure point spectra
2.1. Crystallographic systems
2.2. Model sets
2.3. Homometry
3. Singular continuous spectra
3.1. Thue-Morse sequence
3.2. Generalised Morse sequences
4. Absolutely continuous spectra
4.1. Coin tossing sequence
4.2. Rudin-Shapiro sequence
4.3. Bernoullisation
5. Further directions
5.1. Ledrappier’s model
5.2. Random dimers on the integers
5.3. Renewal processes
5.4. Random clusters and point processes
6. Concluding remarks
References

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MICHAEL BAAKE AND UWE GRIMM

1. Introduction to diffraction theory
The distribution of matter in Euclidean d-space is described by a measure ω on
R , where we assume an infinite system that is homogeneous and in equilibrium. In
most cases, ω will be translation bounded, which means that, for any compact set
K ⊂ Rd , we havedsup |ω|(t + K) < ∞ .

t∈RdMoreover, we assume an amenability property of ω, namely the existence of its autocorrelation measuregω|R ∗ ω|R,
R→∞ vol(BR )γ = γω = ω ⊛ ωe := lim

where BR denotes the open ball of radius R around 0 ∈ Rd and ω|R the restrictiong)of ω to BR . Given a measure µ, its ‘flipped-over’ version µe is defined via µe(g) = µ(edfor g ∈ Cc (R ), the space of continuous (complex-valued) functions g of compact
support, where eg (x) = g(−x). The volume-averaged (or Eberlein) convolution ⊛ isneeded because ω itself is an unbounded measure, so the direct convolution is notdefined. Note that different measures ω can share the same autocorrelation γ.
By construction, the measure γ is positive definite, which means γ(g ∗ eg) ≥ 0for all g ∈ Cc (Rd ). As a consequence, γ is Fourier transformable by general results[22]. The Fourier transform b
γ exists and is a positive measure, called the diffraction
measure of ω. It describes the outcome of kinematic diffraction by ω in the sensethat γb quantifies how much scattering intensity reaches a given volume in d-space.
By the Lebesgue decomposition theorem, relative to Lebesgue measure λ, there is aunique splitting
γb = bγpp + bγsc + bγacof bγ into its pure point part (the Bragg peaks, of which there are at most countablymany), its absolutely continuous part (the diffuse scattering with locally integrable
density relative to λ) and its singular continuous part (which is whatever remains).
The last contribution, if present, is described by a measure that gives no weight tosingle points, but is still concentrated to a set of zero Lebesgue measure.
Systems with bγ = b
γpp are called pure point diffractive. Important examples are
perfect crystals and quasicrystals, such as the icosahedrally symmetric AlMnPd alloythat produces the diffraction image of Figure 1. Mathematical examples of all spectral
types will be discussed below. For general background, we refer to [18], and to [8]in particular. The increasing need for a better understanding of diffuse scattering isevident from [48] and references therein.
It is clear that the diffraction measure is a unique attribute of the structure described by ω (under the mild assumption that γ exists, which is a realistic assumption
from the physical point of view). In contrast, the inverse problem of determining ωMATHEMATICAL DIFFRACTION THEORY

3

Figure 1. Experimental diffraction pattern of an icosahedral AlMnPd
quasicrystal along the fivefold direction (courtesy C. Beeli).

from γb is generally non-unique and a hard problem to solve, both mathematicallyand practically.
For simplicity, we will, whenever possible, explain the different scenarios with Dirac
combs on Z. This means that, given a bi-infinite sequence w = (wn )n∈Z , we consider(1)

ω = w δZ :=X

w(n) δn ,n∈Zwhere δn is the normalised Dirac measure located at n. The weights w(n) = wn are
assumed to be bounded, and we use the notations w(n) and wn interchangeably. A
simple calculation shows that the corresponding autocorrelation (which will exist in

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MICHAEL BAAKE AND UWE GRIMM

all examples discussed here) is then of the form
X
(2)
γ =
η(m) δm ,m∈Zwith the autocorrelation coefficients
N

(3)

X
1
η(m) = lim
w(n) w(n−m).
N →∞ 2N +1
n=−N

In line with our previous remark, we will assume that the limit exists for all m ∈ Z,
which is equivalent to the existence of γ in this case.
Let us now illustrate the possible spectral types by means of important and characteristic examples.
2. Pure point spectraP
The Dirac comb of a general point set S is defined as δS := x∈S δx , with δx thenormalised point measure at x. If Γ ⊂ Rd is a lattice (meaning a discrete co-compact
subgroup of Rd ), the corresponding Dirac comb δΓ itself is Fourier transformable via
the Poisson summation formula (PSF)(4)δbΓ = dens(Γ ) δΓ ∗ ,where Γ ∗ denotes the dual lattice of Γ and dens(Γ ) the density of Γ ; see [19] andreferences therein for details.
2.1. Crystallographic systems. A perfect (infinite) crystal with Γ as its latticeof periods can be described by the measure ω = µ ∗ δΓ , where µ is a suitable finite
measure. A convenient (though not unique) choice for µ is the restriction of ω to afundamental domain of Γ . A simple calculation leads to the autocorrelation
γ = dens(Γ ) (µ ∗ µe ) ∗ δΓ

because δeΓ = δΓ and δΓ ⊛ δΓ = dens(Γ ) δΓ . The Fourier transform of γ exists and
reads
2 2
γb = dens(Γ ) µb δΓ ∗

by an application of the convolution theorem together with the PSF (4). Note that2µb is a uniformly continuous and bounded function that is evaluated only at pointsof the dual lattice Γ ∗ . Different admissible choices for the measure µ lead to different
such functions that agree on all points of Γ ∗ . The measure b
γ is a pure point measure,
as one expects for lattice periodic measures ω.

MATHEMATICAL DIFFRACTION THEORY

5

Figure 2. The silver mean point set Λ of Equation (5) as a√projectionof a strip-shaped subset of the lattice Γ = {(x, x′ ) | x ∈ Z[ 2 ]}. The
shading marks the endpoints of a-type (light grey) and b-type (darkgrey) intervals.

2.2. Model sets. Lattice periodic point sets form a special case of the larger class ofregular model sets [39, 43], which also lead to pure point diffraction measures. Oneof the simplest non-periodic examples in one dimension emerges from the silver mean
substitution rule̺:

a 7→ abab 7→ a√which has inflation multiplier σ = 1 + 2 . The latter is known as the silver mean
and is the Perron-Frobenius eigenvalue of the corresponding substitution matrix ( 21 11 ).
Note that σ is a Pisot-Vijayaraghavan (PV) number.
The natural geometric realisation of this system, starting from a bi-infinite fixed
point of ̺ with legal seed a|a, is built via two intervals of length ratio σ. If a represents
an interval of length σ and b one of length 1, their left endpoints constitute the silver
mean point set
(5)

√ √
√

Λ = x ∈ Z[ 2 ] | x′ ∈ [− 22 , 22 ] ,′the proof of which is not entirely trivial; it is spelled√out in detail in [13].
√ Here, √denotes algebraic conjugation in the quadratic field Q( 2 ), as defined by 2 7→ − 2 .
Equation (5) has a distinctive geometric meaning which is illustrated in Figure 2.

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MICHAEL BAAKE AND UWE GRIMM

The algebraic aspects are encoded in a cut and project scheme (CPS), which westate for the general Euclidean case as follows:
Rd
(6)

∪

1−1

π(L) ←−−−
k

L

π

π

←−− Rd × Rm −−int
−→
∪

L

Rm
∪ dense−−−→ πint (L)⋆−−−−−−−−−−−−−−−−→kL⋆

Here, L is a lattice in Rd+m with certain properties that are expressed via the images
under the canonical projections π and πint . In particular, L = π(L) is a bijective
image of L, while L⋆ = πint (L) is dense in internal space Rm . Due to these properties,
the ⋆ -map x 7→ x⋆ is well-defined on L; see [39]√ for more. In the silver mean example,⋆
we have a CPS with d = m = 1, L = L = Z[ 2 ], and the ⋆ -map is given by algebraic
conjugation, as mentioned above.
In general, a model set for a given CPS is a set of the form(7)
Λ = x ∈ L | x⋆ ∈ W

where W is a relatively compact subset of Rm ; see Equation (5) for the silver meancase. A model set Λ is regular when the boundary ∂W of the window W has zeroLebesgue measure. The entire setting generalises, without significant complications,to locally compact Abelian groups as internal spaces [39, 43]. We will refer to thisfreedom later on, where the internal space will be based on the 2-adic numbers.
Regular model sets are pure point diffractive [31, 43, 19]. This is a substantial
theorem for which three different types of proofs are known. The most common
one is based on the connection to dynamical systems theory [31, 43], another on areformulation via almost periodic measures [19, 28]. An even simpler approach follows
a suggestion by Lagarias and is based on the PSF for the embedding lattice together
with Weyl’s lemma on uniform distribution [13]. The theorem is also constructivein the sense that it provides an explicit and computable formula for the diffraction
measure of δΛ , namelyX|A(k)|2 δk
γ=
b
k∈L⊛

with Fourier module L⊛ = π(L∗ ) and amplitudes
A(k) =

dens(Λ) c
1W (−k ⋆ ) ,vol(W )

where 1W is the characteristic function of the window W . This formula has several
generalisations [43, 19], which we omit for simplicity.
Model sets are widely used to describe and analyse diffraction images such as thatshown in Figure 1. Although real world quasicrystals will usually not be pure point

MATHEMATICAL DIFFRACTION THEORY

7

ξ2
ξ3

ξ⋆

ξ

ξ3

1

⋆

1⋆

ξ2

⋆

Figure 3. Action of the Z-linear ⋆ -map on the generating elementsof the eightfold module L = Z[e2πi/8 ].
diffractive, their average structure is well captured by this approach. In this regard,
quasicrystals behave pretty much like ordinary crystals.
Let us expand on the diffraction formula for the vertex set of the planar AmmannBeenker (or octagonal) tiling [3]. This point set is a regular model set with L = Z[ξ],where ξ = exp(2πi/8) is a primitive eighth root of unity. The ⋆ -map (illustrated in
Figure 3) is defined by ξ 7→ ξ 3 , which is an automorphism of the cyclotomic field Q(ξ),
so that L⋆ = L. The lattice L = {(x, x⋆ ) | x ∈ Z[ξ]} is the Minkowski embedding ofL, which is a scaled copy of Z4 in this case. The standard window is a regular octagon
O of edge length 1, centred at the origin. The model set construction produces thepoint set
ΛAB = x ∈ Z1 + Zξ + Zξ 2 + Zξ 3 | x⋆ ∈ O
and its ⋆ -image Λ⋆AB of Figure 4. The corresponding tiling emerges by connecting allvertices of distance 1 in ΛAB .
The diffraction measure is calculated via the Fourier transform of the characteristic
function 1O . This leads to a dense (but countable) set of Bragg peaks whose intensities
are locally summable. This means that, in any compact region, there are only finitely
many peaks above any given positive threshold. A precise explicit calculation caneasily be performed by one of the standard computer algebra packages. Figure 5
shows the result for a central patch of the Ammann-Beenker diffraction, with cutoff
at 1/1000 of the central intensity. Here, a Bragg peak is represented by a small disk
whose area is proportional to the intensity and whose centre is the position of the
Bragg peak.
A related and very interesting question concerns the spectral type of systems thatare defined by substitutions; see [41] for background. More recently, practically useful
tests for pure pointedness of substitution systems have been developed, compare [2]and references given there, but the famous Pisot substitution conjecture and its

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MICHAEL BAAKE AND UWE GRIMM

Figure 4. Patch of the Ammann-Beenker tiling and point set ΛAB(left) and its lift to internal space (right).
higher dimensional generalisations still remain a mystery. We will not discuss thispoint of view in what follows, even though many of our examples will be defined bysubstitutions rules.
Let us close the paragraph with a brief general comment. A translation boundedmeasure ω also defines a dynamical system under the translation action of Rd ; see[43, 16, 17] for background. If ω is pure point diffractive, the corresponding dynamical
spectrum is pure point as well (the converse also being true). This equivalence is wellunderstood by now [36, 43, 16, 37], but does not extend to general systems with
continuous spectral components [25, 7], as we will see later on.
2.3. Homometry. As mentioned previously, different measures may possess thesame autocorrelation and hence the same diffraction. This phenomenon is calledhomometry [40]. Clearly, δt ∗ ω (with t ∈ Rd ) as well as ωe have the same diffractionas ω, but non-uniqueness is generally not exhausted by this. Let us illustrate how it
appears already among pure point diffractive systems.
The simplest situation emerges for periodic Dirac combs on Z with rational weights.
As an example, Grünbaum and Moore [30] constructed homometric Dirac combs of
the form
5X(8)
ω = δ6Z ∗
cj δj
j=0

MATHEMATICAL DIFFRACTION THEORY

9

Figure 5. Computed diffraction image of the Ammann-Beenker pointset ΛAB ; see text for details.
with integer weights cj , which are thus 6-periodic. The two choices of Table 1 leadto the same autocorrelation. Even worse, these two cases have the same correlation
functions up to 5th order, and differ only in higher orders. Nevertheless, the two
Dirac combs are substantially different. Note that the diffraction measure, which is
supported on Z/6, shows systematic extinction (the intensity vanishes on all points
Table 1. Integer weights cj for the two homometric Dirac combs built
from Equation (8); see [30] for details.
j 0 1 2 3 4 5
cj 11 25 42 45 31 14
cj 10 21 39 46 35 17

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MICHAEL BAAKE AND UWE GRIMM

Figure 6. Two polyominoes (left) and their common covariogram(right). The small dots mark the origin, see text for further details.
of the form k = ℓ/6 with k ≡ 2, 3 or 4 mod 6). Such extinctions are always anindication for non-trivial homometry of pure point diffractive systems.
Let us go one step beyond by showing a pair of homometric model sets. We use the
CPS of the Ammann-Beenker tiling from above, but replace the octagonal window
by one of the two polyominoes shown in Figure 6, where we follow [9]. The Dirac
comb δΛ of the corresponding model set Λ with window W has an autocorrelation of
the formX(9)γΛ =η(z) δz ,z∈Λ−Λwhere Λ−Λ is locally finite because Λ is a model set. The autocorrelation coefficientsin (9) are given by
vol W ∩ (W − z ⋆ )(10)η(z) = dens(Λ)= dens(L) cvgW (z ⋆ ) .
vol(W )

Here, cvgW is the covariogram of W , defined by
cvgW (x) = vol W ∩ (x + W ) = 1W ∗ 1−W (x) .

This function is symmetric under reflection in the origin, and one also has the relations
cvgt+W = cvgW for arbitrary t ∈ Rd as well as cvg(−W ) = cvgW . The covariogram forour two polyominoes is illustrated as a contour plot in Figure 6. The polyominoes
in the same figure are shown with a shifted overlay structure, which can be used by
the reader to check the claimed homometry (for the displayed shift) on the basis of
Equations (9) and (10).
The two model sets constructed this way differ on positions of positive density.
Depending on the length scale of the windows, they may or may not be locally equivalent via a mutual local derivation (MLD) rule [4], but they are always homometric.
Further details are discussed in [9, 29]. If one has access to correlation functions of

MATHEMATICAL DIFFRACTION THEORY

1

11

F(x)

3/4

1/2

1/4

0

1/3

2/3

1

x

Figure 7. Illustration of the construction of the middle-thirds Cantor
set C (inlay) and the distribution function F for the corresponding
probability measure on C.
higher order, a distinction is possible. A rather general result in this direction wasrecently derived in [24].

3. Singular continuous spectra
The probably best known singular continuous measure is the one that emerges fromthe middle-thirds Cantor set construction. Its distribution function F is shown in
Figure 7, which is widely known as the Devil’s staircase. This function is continuousand non-decreasing, but constant almost everywhere. More precisely, the underlying
measure µ = dF is concentrated on the Cantor set C, which is an uncountable setof zero Lebesgue measure. The (positive) measure µ is singular continuous, withµ({x}) = 0 for all x ∈ [0, 1] and µ(C) = 1.
3.1. Thue-Morse sequence. Let us now discuss a classic example from the theoryof substitution systems that leads to a singular continuous diffraction measure withrather distinctive features in comparison with the Cantor measure. This example has
a long history, which can be extracted from [47, 38, 33, 1]. We confine ourselves to abrief summary of the results, and refer to [10] and references therein for proofs anddetails.

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MICHAEL BAAKE AND UWE GRIMM

The classic Thue-Morse (TM) sequence can be defined via the one-sided fixed pointv = v0 v1 v2 . . . with v0 = 1 of the primitive substitution rule
1 7→ 11̄
1̄ 7→ 1̄1

̺:on the binary alphabet {1, 1̄}. It is the limit (in the obvious product topology) of
the iteration sequence
̺

̺

̺

̺

1 7−→ 11̄ 7−→ 11̄1̄1 7−→ 11̄1̄11̄111̄ 7−→ . . . −→ v = ̺(v) = v0 v1 v2 v3 . . .
and has a number of distinctive properties [1, 41], for instance• vi = (−1)sum of the binary digits of i
• v2i = vi and v2i+1 = v̄i• v = v0 v2 v4 . . . and v̄ = v1 v3 v5 . . .
• v is (strongly) cube-free.

A two-sided sequence w can be defined by(vi ,for i ≥ 0,wi =v−i−1 , for i < 0,which is a fixed point of ̺2 , because the seed w−1 |w0 = 1|1 is a legal word (it occursin ̺3 (1)) and w = ̺2 (w). The (discrete) hull X = XTM of the TM substitution isthe closure of the orbit of w under the shift action, which is compact. The orbit of
any of its members is dense in X. We thus have a topological dynamical system(X, Z) that is minimal. When equipped with the standard Borel σ-algebra, thesystem admits a unique shift-invariant probability measure, so that the corresponding
measure theoretic dynamical system is strictly ergodic [33, 41].
Any given w ∈ X is mapped to a signed Dirac comb (and hence to a translation
bounded measure) ω viaXω =wn δn .
n∈Z

We inherit unique ergodicity, and thus obtain an autocorrelation of the form (2) withcoefficients η(m) as in (3). Due to the nature of the fixed point w, an alternative way
to express the coefficients is
N −1

1 X
vn vn+m
N →∞ Nn=0

η(m) = lim

for m ≥ 0 together with η(−m) = η(m). It is clear that η(0) = 1, and the scaling
relations of v lead to the recursions
1(11)
η(2m) = η(m) and η(2m+1) = − η(m) + η(m+1)
2which are valid for all m ∈ Z. In particular, the second relation, used with m = 0,
implies η(1) = − 31 , which can also be calculated directly.

MATHEMATICAL DIFFRACTION THEORY

13

Since ω is supported on Z, the diffraction measure γb is of the form b
γ = µ ∗ δZ by
an application of [5, Thm. 1], whereZ 1µ=bγ [0,1) together with η(m) =e2πimy dµ(y) ,0the latter due to the Herglotz-Bochner theorem. One can now analyse the spectral type of bγ via that of the finite measure µ, where we follow [33]. Defining
2
PN
Σ(N) = m=−N η(m) , a two-step calculation with the recursion (11) establishes
the inequality Σ(4N) ≤ 23 Σ(2N) for all N ∈ N. This implies limN →∞ Σ(N)/N = 0,
wherefore Wiener’s criterion [47] tells us that µ is a continuous measure, so that bγcannot have any pure point component.
Let us now define the distribution function F by F (x) = µ [0, x] for x ∈ [0, 1],
which is a continuous function that defines a Riemann-Stieltjes measure, so thatdF = µ. The recursion relation for η now implies [33] the functional relations
 1= − cos(πx)
dF (x) ,dF x2 ± dF x+1
2
which have to be satisfied by the ac and sc parts of F separately, because µac ⊥ µsc
in the measure theoretic sense. Therefore, defining
Z 1
ηac (m) =e2πimx dFac (x) ,0

we know that the coefficients ηac (m) must satisfy the same recursions (11) as η(m),possibly with a different initial condition ηac (0). The Riemann-Lebesgue lemma states
limm→±∞ ηac (m) = 0, with is only compatible with ηac (0) = 0, and hence ηac ≡ 0.
This means Fac = 0 by the Fourier uniqueness theorem, wherefore µ and hence bγ(neither of which is the zero measure) are purely singular continuous. The resulting
distribution function is illustrated in Figure 8. It was calculated by means of thequickly converging Volterra iterationZ
1 2x
1 − cos(πy) Fn′ (y) dy
withF0 (x) = x .
Fn+1 (x) =2 0

In contrast to the Devil’s staircase, the TM function is strictly increasing, whichmeans that there is no plateau (which would indicate a gap in the support of bγ ); see[10] and references therein for details and further properties of F .
Despite the above result, the TM sequence is closely related to the period doubling
sequence, via the (continuous) block map
(12)

φ:

11̄, 1̄1 7→ a ,11, 1̄1̄ 7→ b ,which defines an exact 2-to-1 surjection from the hull XTM to Xpd , where the latteris the hull of the period doubling substitution defined by
(13)

̺pd :

a 7→ ab ,b 7→ aa .

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MICHAEL BAAKE AND UWE GRIMM

1

0.5

0
0

0.5

1

Figure 8. The strictly increasing distribution function F of the ThueMorse diffraction measure on the unit interval.
Viewed as topological dynamical systems, this means that (Xpd , Z) is a factor of(XTM , Z). Since both are strictly ergodic, this extends to the corresponding measure
theoretic dynamical systems. The period doubling sequence can be described asa regular model set with a 2-adic internal space [20, 19] and is thus pure point
diffractive. This pairing also explains a phenomenon observed in [25], as the missing
part of the dynamical spectrum of the TM system is recovered via the diffraction
measure of Xpd .
3.2. Generalised Morse sequences. The above example can be generalised to thefamily̺:

1 7→ 1k 1̄ℓ1̄ 7→ 1̄k 1ℓ

MATHEMATICAL DIFFRACTION THEORY

1

1

0.5

0.5

0

15

0
0

0.5

1

0

0.5

1

Figure 9. The continuous and strictly increasing distribution functions of for the generalised Morse sequences with parameters (2, 1)(left) and (5, 1) (right).
with k, ℓ ∈ N, inspired by [34]. They define a class of systems which we will referto as the gTM systems. All display purely singular continuous diffraction, which
follows from completely analogous arguments [12]. The entire analysis is based on
the structure of the autocorrelation, which reads γ = η δZ with η(0) = 1 and therecursion relations

1
αk,ℓ,r η(m) + αk,ℓ,k+ℓ−r η(m + 1) ,η (k + ℓ)m + r =k+ℓwith αk,ℓ,r = k + ℓ − r − 2 min(k, ℓ, r, k + ℓ − r). They are valid for all m ∈ Z and
0 ≤ r < k + ℓ. In particular, one has η (k + ℓ)m = η(m) for m ∈ Z.
Given k, ℓ ∈ Z, the distribution function F is defined by F (x) = b
γ ([0, x]) for0 ≤ x < 1, which extends to x ∈ R via F (x + 1) = 1 + F (x). It is also skewsymmetric (F (−x) = −F (x)) and thus satisfies F (q) = q for all q ∈ 21 Z. The
continuous function F possesses the uniformly converging series expansion [12]X η(m)sin(2πmx) .
F (x) = x +mπm≥1
Two further examples are shown in Figure 9.
Another analogy with the TM system is that the block map φ of Equation (12) canstill be used to induce a matching family of generalised period doubling sequences.
They are defined by the primitive substitution rules
̺′ :a 7→ bk−1 abℓ−1 b
b 7→ bk−1 abℓ−1 a

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MICHAEL BAAKE AND UWE GRIMM

that are all pure point diffractive. The latter claim can most easily be seen from theletter coincidence (in the sense of Dekking [23]) at the kth position of the images.
As before, this factor explores the pure point part of the dynamical spectrum of thegTM sequence.
4. Absolutely continuous spectra
The appearance of absolutely continuous diffraction spectra is usually seen as
an indicator for randomness in the structure. Though this is perhaps genericallytrue, there are also prominent deterministic sequences with such spectra, such as the
Rudin-Shapiro sequence. In general, within the realm of random structures, one canonly expect almost sure convergence results. In other words, most statements become
measure theoretic in nature, though they are still completely rigorous.
4.1. Coin tossing sequence. The simplest example emerges from repeated cointossing. Here, one obtains sequences w ∈ {±1}Z (for instance with 1 for ‘head’ and−1 for ‘tail’) which may be considered as the outcome of an eternal coin tosser. In
more modern (and slightly more general) terminology, one considers a family Wn n∈Zof independent and identically distributed (i.i.d.) random variables with values in{±1} and probabilities p (for 1) and 1 − p (for −1). The ensemble of all possible
realisations is XB = {±1}Z , which is equipped with a probability measure µB thatemerges from the elementary probabilities via independence. This gives a measure
theoretic dynamical system that is called the Bernoulli shift [46]. It has (metric)
entropy H(p) = −p log(p) − (1−p) log(1−p).
A random sequence W leads to a Dirac comb ω = W δZ , which is now a translation
bounded random measure with support Z. Its autocorrelation, if it exists, is of theform γB = ηB δZ with(N1,
m = 0,
1 X(a.s.)Wn Wn+m =ηB (m) := lim
N →∞ 2N+1(2p−1)2, m 6= 0.
n=−NHere, the convergence is almost sure by the strong law of large numbers (SLLN) [26],which means that this is the result for µB -almost all elements of {±1}Z . Note thatthe use of the SLLN can also be replaced by an application of Birkhoff’s ergodic
theorem, because the Bernoulli shift is ergodic [46]. The corresponding diffraction
measure reads(a.s.)2γc
B = (2p − 1) δZ + 4p(1 − p) λ ,which follows from γB by an application of the PSF (4) together with δb0 = λ. For
the fair coin (p = 12 ), this simplifies to γc
B = λ, which is thus our first example of apurely absolutely continuous diffraction measure.

MATHEMATICAL DIFFRACTION THEORY

17

Figure 10. Illustration of the binary Rudin-Shapiro sequence.
4.2. Rudin-Shapiro sequence. Let us contrast the coin tossing sequence with adeterministic example that derives from [42, 45]. This sequence was originally constructed to show that the absence of pair correlations does not imply the presence ofrandomness. This has interesting consequences in diffraction theory, as pointed out
in [32]. The modern formulation of the system is based on the substitution
̺:a 7→ ac,b 7→ dc,

c 7→ ab,d 7→ db .

Since b|a is a legal seed (it occurs in ̺2 (b)), one can construct a bi-infinite sequence
u by the usual iteration procedure as
̺2

̺2

b|a 7−→ dbab|acab 7−→ . . . −→ u = ̺2 (u) ,where convergence is in the standard product topology. The hull (orbit closure) ofu defines the quaternary Rudin-Shapiro system. Its reduction to a binary system isachieved by the mapping
ϕ:a, c 7→ 1 ,b, d 7→ 1̄ ,and the orbit closure of w := ϕ(u) = . . . 1̄1̄11̄|1111̄ . . . defines the hull XRS of thebinary Rudin-Shapiro system. The sequence w is illustrated in Figure 10. For theequivalent description as a weighted Dirac comb on Z, we again use the identificationof 1̄ with −1.
An alternative description of w uses the initial conditions w(−1) = −1, w(0) = 1
together with the recursion(
w(n),
for ℓ ∈ {0, 1},(14)w(4n + ℓ) =n+ℓ(−1) w(n), for ℓ ∈ {2, 3}.
The autocorrelation of the corresponding weighted Dirac comb ωRS exists and turns
out to be γRS = δ0 . To prove this, one defines the coefficients
η(m)
ϑ(m)




N
X
1
1
w(n) w(n + m):= limN →∞ 2N+1(−1)n .
n=−N

An application of Birkhoff’s ergodic theorem to the quaternary Rudin-Shapiro system (which is strictly ergodic [41]) establishes the existence of all these limits. The

18

MICHAEL BAAKE AND UWE GRIMM

recursion (14) for w now implies the recursion relations [12]m

η(4m) = 1+(−1)
η(m),
2
m

η(4m+2) = 0,m

η(4m+1) = 1−(−1)η(m) + (−1)ϑ(m) − 41 ϑ(m+1),
4
4
m

m

η(m+1) − (−1)ϑ(m) + 14 ϑ(m+1),
η(4m+3) = 1+(−1)44
ϑ(4m) = 0,m

ϑ(4m+2) = (−1)ϑ(m) + 12 ϑ(m+1),2mm

η(m) − (−1)ϑ(m) + 14 ϑ(m+1),
ϑ(4m+1) = 1−(−1)
4
4
m

m

ϑ(4m+3) = − 1+(−1)η(m+1) − (−1)ϑ(m) + 14 ϑ(m+1).
4
4
with initial conditions η(0) = 1 and ϑ(0) = 0. This system has the unique solutionϑ ≡ 0 together with η(m) = δm,0 , hence γRS = δ0 .
The diffraction measure for the binary Rudin-Shapiro system is thus given byγRS = λ, which coincides with that of the coin tossing sequence for p = 21 ; seebalso [32]. These two examples are thus homometric, despite the fact that one isdeterministic (with entropy 0) while the other is stochastic (with entropy log(2)); see[11, 12] for further details and discussions.
4.3. Bernoullisation. The homometry between the Dirac combs of the RudinShapiro and the balanced coin tossing sequence raises the question how ‘bad’ thenon-uniqueness of the inverse problem in this case really is. A partial answer (tothe negative) can be given by means of the ‘Bernoullisation’ procedure which wasintroduced in [29, 11].
Starting from a uniquely ergodic sequence S ∈ {±1}Z , its Dirac comb ωS = S δZ
possesses a unique autocorrelation γS . Let us now consider the random Dirac comb
X
Sn Wn δn ,
ω=
n∈Z



where Wn n∈Z is once again an i.i.d. family of random variables with values in {±1}and probabilities p and 1 − p. Another application of the SLLN shows that ω almost
surely has the autocorrelation(a.s.)

γ = (2p − 1)2 γS + 4p(1 − p) δ0 .
If S is the binary Rudin-Shapiro sequence, which is uniquely ergodic, a short calculation reveals thatγS = γRS = δ0in this case, irrespective of the value of the parameter p ∈ [0, 1]. This way, weconstructed a one-parameter family of homometric (or isospectral) structures whose
entropy varies continuously between 0 and log(2). The conclusion is that kinematic
diffraction alone cannot distinguish order from disorder here [11].

MATHEMATICAL DIFFRACTION THEORY

19

5. Further directions
After our brief discourse on the different spectral types of diffraction measures (bymeans of some paradigmatic examples), this section aims at indicating some more
recent developments, which will again be explained informally by means of selected
examples.
5.1. Ledrappier’s model. Let us consider a prominent example of algebraic origin
in the plane which is due to Ledrappier [35]. It is defined as2
XL = w ∈ {±1}Z | wx wx+e1 wx+e2 = 1 for all x ∈ Z2 ,where e1 and e2 denote the standard Euclidean basis vectors in the plane. XL is2
a closed subset of the full shift {±1}Z and hence compact. It is also an Abeliangroup (under pointwise multiplication in our formulation, which follows [21]). Asa dynamical system, it is thus equipped with the corresponding Haar measure µL,
which is positive and normalised so that µL (XL ) = 1. Obviously, the system has no
entropy, because the knowledge of a configuration along one horizontal line determines
everything above it. However, it is clearly not deterministic. In fact, essentially along
any given lattice direction, it looks like a one-dimensional Bernoulli system [35]. It
is thus said to have rank 1 entropy, which means that the number of circular patchesof a given size grows exponentially with its diameter, but not with its area.
Given an element w ∈ XL , the corresponding Dirac comb
X
wx δx
ω =
x∈Z2

possesses µL -almost surely the autocorrelation γ and the diffraction measure b
γ givenby [21]γ = δ0 and γb = λ .

The system is thus homometric with the two-dimensional Bernoulli system with p = 21
(coin tossing on Z2 ), and also with the direct product of two binary Rudin-Shapiro
sequences. The similarity with the Bernoulli system goes a lot further, in the sensethat also other correlation functions agree, although the systems differ for certain
3-point correlations; see [21] for details.
This system is meant to indicate that higher-dimensional symbolic dynamics isgood for a surprise, as is well-known from [44]. It is thus clear that the inverse problem
becomes more complicated. Another famous example is the (×2, ×3) dynamicalsystem, which shares almost all correlation functions with a Bernoulli system with
continuous degree of freedom [21].
5.2. Random dimers on the integers. Back to one dimension, let us briefly describe a system that was recently suggested by van Enter. First, consider Z as aclose-packed arrangement of ‘dimers’ (pairs of neighbours), hence without gaps or

20

MICHAEL BAAKE AND UWE GRIMM

overlaps. There are two possibilities to do so. Next, give each pair a random orientation by decorating it with either (+, −) or (−, +), with equal probability. Identifying± with ±1, this defines the closed (and hence compact) set
X = w ∈ {±1}Z | M(w) ⊂ 2Z or M(w) ⊂ 2Z + 1 ,where M(w) := {i ∈ Z | wi = wi+1 }. Note that M(w) can be empty, which happens
precisely for the two periodic sequences . . .+−|+− . . . and . . .−+|−+ . . .. One has aninvariant measure on X that emerges from the stochastic process via the probabilityof the possible finite patches (which define the generating cylinder sets as usual).
Turning a configuration w ∈ X into a signed Dirac comb with weights wi ∈ {±1}as before, another exercise with the SLLN shows that its autocorrelation almost
surely exists. It is not difficult to derive [7] that γ = δ0 − 21 (δ1 + δ−1 ), so that thecorresponding diffraction measure isγcw = 1 − cos(2πk) λ .

This is another example of a purely absolutely continuous diffraction measure. The
Radon-Nikodym density relative to λ is written as a function of k. In contrast, thedynamical spectrum of this system contains eigenvalues, wherefore this is an analogueof the comparison between the Thue-Morse and period doubling systems [25], thistime in the presence of absolutely continuous spectra.
This difference can be rectified by a block map that is very similar to the map φencountered in (12). Defining ui = −wi wi+1 for i ∈ Z maps w to a new sequence u,which almost surely has the diffraction11
δZ/2 + λ4
2of mixed type. In particular, as in the Thue-Morse case, it displays the entire pointpart of the original dynamical spectrum.
cγu =5.3. Renewal processes. A large and interesting class of processes in one dimension
can be described as a renewal process [6, 15]. Here, one starts from a probability
measure µ on R+ (the positive real line) and considers a machine that moves atconstant speed along the real line and drops a point on the line with a waiting timethat is distributed according to µ. Whenever this happens, the internal clock is reset
and the process resumes. Let us (for simplicity) assume that both the velocity of themachine and the expectation value of µ are 1, so that we end up with realisations
that are, almost surely, point sets in R of density 1 (after we let the starting point of
the machine move to −∞).
Clearly, the resulting process is stationary and can thus be analysed by considering
all realisations which contain the origin. Moreover, there is a clear (distributional)
symmetry around the origin, so that we can determine the corresponding autocorrelation γ of almost all realisations from studying what happens to the right of 0.

MATHEMATICAL DIFFRACTION THEORY

21

Indeed, if we want to know the frequency per unit length of the occurrence of two
points at distance x (or the corresponding density), we need to sum the contributions
that x is the first point after 0, the second point, the third, and so on. In other words,
we almost surely obtain the autocorrelation(15)

γ = δ0 + ν + νewith ν = µ + µ ∗ µ + µ ∗ µ ∗ µ + . . ., where the proper convergence of the sum of
iterated convolutions follows from [6, Lemma 4]. Note that the point measure at 0
simply reflects that the almost sure density of the resulting point set is 1. Indeed,ν is a translation bounded positive measure, and satisfies the renewal relations (see[27, Ch. XI.9] or [6, Prop. 1] for a proof)(16)ν = µ+µ∗νand(1 − µb ) νb = µb,where µb is a uniformly continuous and bounded function on R. Note that the second
equation emerges from the first by Fourier transform, but has been rearranged toindicate why the set S = {k | µb(k) = 1} is important. In this setting, the measure γof (15) is both positive and positive definite.
Based on the structure of the support of the underlying probability measure µ,
one can determine the diffraction of the renewal process as follows. Let µ be aprobability measure on R+ with mean 1, and assume that a moment of µ of order
1 + ε exists for some ε > 0 (we refer to [6] for details on this condition). Then, thepoint sets obtained from the stationary renewal process based on µ almost surely
have a diffraction measure of the form
γ = bbγpp + (1 − h) λ,where h is a locally integrable function on R that is continuous except for at mostcountably many points (namely those of the set S). It is given by
2 |bµ(k)|2 − Re(bµ(k)).
h(k) =|1 − µb(k)|2Moreover, the pure point part reads(
δ0 ,if supp(µ) is not a subset of a lattice,γbpp =
δZ/b , if bZ is the coarsest lattice that contains supp(µ).

Proofs can be found in [6, 15].
The renewal process is a versatile method to produce interesting point sets on theline. These include random tilings with finitely many intervals (which are Delone sets)
as well as the homogeneous Poisson process on the line (where µ is the exponential
distribution with mean 1); see [6, Sec. 3] for explicit examples and applications.

22

MICHAEL BAAKE AND UWE GRIMM

5.4. Random clusters and point processes. Let us continue by considering theinfluence of randomness on the diffraction of point sets and certain structures derived
from them in Euclidean spaces of arbitrary dimension. Here, we start from a single
point set Λ ⊂ Rd , which is then randomly modified by replacing each point by a(possibly complex) finite random cluster. This situation is still manageable (via the
SLLN) when Λ is sufficiently ‘nice’, for instance if Λ is of finite local complexity andPpossesses an autocorrelation, which is then of the form γ = z∈Λ−Λ η(z) δz . Moregenerally, one can analyse this situation in the setting of stationary ergodic point
processes [28, 6], which treats almost all of their realisations at once and permits alarger generality for the sets Λ, though the clusters will then be restricted to positiveor signed measures.
Given such a point set Λ, its (deterministic) Dirac comb δΛ is turned into a random
Dirac comb
X
(Ω)
δΛ =
Ωx ∗ δxx∈Λ

by means of the i.i.d. family of random measure (Ωx )x∈Λ with common law Q and
representing random variable Ω. Here, we
assumethattheexpectation
E
|Ω|
is
Q

d 2
a finite measure, and that EQ (|Ω|(R )) < ∞. Under some mild (but somewhat
technical) conditions [6], one now obtains the autocorrelation
(a.s.)γ (Ω) = EQ (Ω) ∗ E^
Q (Ω) ∗ γe − EQ (Ω) ∗ E^
+ dens(Λ) EQ (Ω ∗ Ω)
Q (Ω) ∗ δ0and hence the diffraction(a.s.)

γb(Ω) =

b 2 · γb + dens(Λ) EQ (|Ω|b 2 ) − |EQ (Ω)|
b 2 · λ.
EQ (Ω)The diffraction of the modified structure emerges from the original one by a modulation of γb and the addition of an absolutely continuous contribution, which in essenceis the Fourier transform of the covariance of the representing random cluster Ω.
This approach comprises a wide range of models, including the random weight
and the random displacement model as well as decorations by random clusters. The
next level of generality replaces the deterministic set Λ by a general ergodic point
process [28]. This way, both the underlying set (core process) and the modification(cluster process) are described in terms of stochastic processes. This point of view is
a promising starting point for further investigations.
6. Concluding remarks
Our informal exposition was meant to demonstrate that mathematical diffraction
theory provides useful tools for the analysis of deterministic and random systems,
both for practical applications in crystallography and materials science and for theoretical questions in harmonic analysis and dynamical systems theory. While theMATHEMATICAL DIFFRACTION THEORY

23

majority of the crystallographic literature concentrates on the pure point case, wehave shown that also continuous spectra are explicitly accessible in relevant cases, andcertainly deserve more attention from this point of view. Merging methods from harmonic analysis and dynamical systems with well-established procedures from point
process theory might be a good path to proceed.
The inverse problem of structure determination is already a formidable problem
in the realm of pure point diffractive systems, due to the existence of non-triviallyhomometric structures. As we have shown above, the inverse problem becomes significantly more involved in the presence of disorder, including the potential insensitivity
to quantities such as entropy. Although there is quite some knowledge about thisproblem in the point process community, it is fair to say that solutions to the inverse
problem or satisfactory classifications of homometry classes are not in sight.
The setting is by no means restricted to lattice systems, which were mainly chosen for ease of presentation and concreteness of results. Also extensions to higher
dimensions are possible, where one has to expect new phenomena (such as the lower
rank entropy) that further complicate the picture. Although the general theory ofpoint processes is highly developed, the treatment of stochastic systems with interaction becomes difficult as soon as concrete results are desired. This is already thecase for equilibrium systems, as they require the full machinery of Gibbs measures.
Their analysis from the point of view of mathematical diffraction theory is still in its
infancy, although examples such as the Ising lattice gas show (see [14] and references
therein) that explicit results are possible.
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Fakultät für Mathematik, Universität Bielefeld,
Postfach 100131, 33501 Bielefeld, Germany
Department of Mathematics and Statistics, The Open University,
Walton Hall, Milton Keynes MK7 6AA, UK