Applications of the notion of model set

This section is devoted to the application of the notion of model set to the calculation of the rate of injectivity of a finite number of matrices.

We take advantage of the rational independence between the matrices of a generic sequence to state a generalization in arbitrary times of the geometric formulas of Sec-tion8.2. The end of the present section is devoted to the proof of the main theorem of this chapter (Theorem8.24).

Let us summarize the different notations we will use throughout this section. We will denote by 0^k the origin of the spaceR^k, and W^k =]−1/2,1/2]^nk (unless otherwise stated). In this section, we will denote D_c(E) the density of a “continuous” set E⊂Rⁿ, defined as (when the limit exists)

D_c(E) = lim

R→+∞

Leb(B_R∩E) Leb(B_R) ,

while for a discrete set E⊂Rⁿ, the notation D_d(E) will indicate the discrete density of E, defined as (when the limit exists)

D_d(E) = lim These quantities will be linked together during this section.

8.7.1 A geometric viewpoint to compute the rate of injectivity in arbitrary

Here, we suppose that the set p₁(Λ_k) is dense (thus, equidistributed) in the image set imp₁ (note that this condition is generic among the sequences of invertible linear maps). In particular, the set{p₂(γ)|γ∈Λ_k}is equidistributed in the window W^k.

The following property makes the link between the density ofΓ_k — that is, the rate of injectivity of A₁,· · ·,A_k — and the density of the union of unit cubes centred on the points of the latticeΛ_k.

Proposition 8.40. For a generic sequence of matrices(A_k)_k ofSL₂(R), we have D_d(Γ_k) = D_c

Of course, this proposition generalizes Proposition 8.11to an arbitrary number of matrices A₁,· · ·,A_k.

Remark8.41. The density on the left of the equality is the density of a discrete set (that is, with respect to counting measure), whereas the density on the right of the equality is that of a continuous set (that is, with respect to Lebesgue measure). The two notions coincide when we consider discrete sets as sums of Dirac masses.

This proposition leads to the definition of the mean rate of injectivity in timek.

Definition 8.42. Let (A_k)_k≥1 be a sequence of matrices of SL_n(R). The mean rate of injectivity in timekof this sequence is defined by

τ^k(A₁,· · ·,A_k) = D_c

W^k+1+Λ_k .

Remark8.43. As in Section8.2for the rate of injectivity in time 1, Proposition8.40 as-serts that for a generic sequence of matrices, the rate of injectivityτ^kin timekcoincides with the mean rate of injectivityτ^k, which is continuous and piecewise polynomial of degree≤nkin the coefficients of the matrix.

Remark8.44. The formula of Proposition8.40could be used to compute numerically the mean rate of injectivity in time k of a sequence of matrices: it is much faster to compute the area of a finite number of intersections of cubes (in fact, a small number) than to compute the cardinalities of the images of a big set [−R,R]ⁿ×Zⁿ.

Proof of Proposition8.40. We want to determine the proportion of integral pointsx∈Zⁿ such that there existsλ∈Λ_ksuch thatp₂(λ) =xandp₁(λ)∈W^k. This depends only on the position of the setp₁

p₂⁻¹(x)∩Λ_k

. More precisely, if we set

Me_A1,···,A_k=

only differ by a translation of the form (0^(k−1)n,−x) (see Figure8.15). Recall that our aim is to decide whether there

To prove this equidistribution, we compute the inverse matrix ofMe_A1,···,A_k:

Me⁻_A¹

ofZⁿon the canonical torusR^nk/Z^nk. But this action is trivially ergodic (even in restric-tion to the first coordinate) when the sequence of matrices is generic.

As in the case of the time 1 (Proposition8.12), there is also a geometric method to compute the frequency of a difference inΓ_k.

W^k

Proposition 8.45. For a generic sequence of matrices(A_k)_k, we have ρ_Γk(v) = Leb

Proof of Proposition8.45. To computeρ_Γ

k(v), we want to know for which proportion of x∈Zⁿsuch thatx∈Γ_k, we havex+v∈Γ_k, that is, knowing thatp₁

are equidistributed in W^k(see the proof of Proposition8.40), the proportion of such pointsx is equal to the area of the intersection

W^k+1+Λ_k

∩ W^k+1+ (0^kn, v)

.

As in the case of time 1 (see Section8.2), an argument of equidistribution combined with Proposition7.27allows to deduce Proposition8.40from Proposition8.45.

There is also a dual method to compute the rate of injectivity of a sequence of ma-trices: we defineψ:R^nk→Rby

ψ= X

λ∈Λ_k

1_Wk+λ,

and obtain the following formula (which is a generalization of Proposition8.18).

Proposition 8.46. For a generic sequence of matrices(A_k)_k, we have D_d(Γ_k) =

Z

B_1/2

1

ψ(λ) d Leb(λ).

The proof of this proposition is similar to that of Proposition8.18page159(see in particular Lemma8.19).

Recall the problem raised by Theorem8.24: we want to makeτ^k as small as possi-ble. By an argument of equidistribution, generically, it is equivalent to make the mean

rate of injectivity τ^k as small as possible whenk goes to infinity, by perturbing every matrix in SL_n(R) of at mostδ>0 (fixed once for all). In the framework of model sets, Theorem 8.24is motivated by the phenomenon of concentration of the measure on a neighbourhood of the boundary of the cubes in high dimension.

Proposition 8.47. LetW^kbe the infinite ball of radius1/2inR^kandv^kthe vector(1,· · ·,1)∈ R^k. Then, for everyε,δ>0, there existsk₀∈N^∗such that for everyk≥k₀, we haveLeb

W^k∩ (W^k+δv^k)

<ε.

Applying Hajós theorem (Theorem8.21), it is easy to see when the density ofΓ_k is equal to 1: combining this theorem with Proposition 8.40, we obtain that this occurs if and only if the lattice given by the matrix M_A1,···,A_k satisfies the conclusions of Hajós theorem⁹. The heuristic suggested by the phenomenon of concentration of the measure is that if we perturb “randomly” any sequence of matrices, we will go “far away” from the lattices satisfying Hajós theorem and then the rate of injectivity will be close to 0.

Remark8.48. The kind of questions addressed by Hajós theorem are in general quite delicate. For example, we can wonder what happens if we do not suppose that the centres of the cubes form a lattice of Rⁿ. O. H. Keller conjectured in [Kel30] that the conclusion of Hajós theorem is still true under this weaker hypothesis. This conjecture was proven to be true forn≤6 by O. Perron in [Per40a,Per40b], but remained open in higher dimension until 1992, when J. C. Lagarias and P. W. Shor proved in [LS92] that Keller’s conjecture is false forn≥10 (this result was later improved by [Mac02] which shows that it is false as soon asn≥8; the casen= 7 is still open).

8.7.2 New proof that the asymptotic rate of injectivity is generically smaller than1/2

As a first application of the concept of model set, we give a new proof of the fact that rate of injectivity of a generic sequence of SL_n(R) is smaller than 1/2. To begin with, we give a lemma estimating the sizes of intersections of cubes when the rate is bigger than 1/2 (which is the geometric counterpart of Lemma8.29).

Lemma 8.49. Let W^k =]−1/2,1/2]^k and Λ⊂ R^k be a lattice with covolume 1 such that D_c(W^k+Λ)≥1/2. Then, for everyv∈R^k, we have

D_c

(W^k+Λ+v)∩(W^k+Λ)

≥2D_c(W^k+Λ)−1.

Proof of Lemma8.49. We first remark that D_c(Λ+ W^k) is equal to the volume of the pro-jection of W^k on the quotient spaceR^k/Λ. For everyv∈R^k, the projection of W^k+v on R^k/Λhas the same volume; as this volume is greater than 1/2, and as the covolume of Λis 1, the projections of W^k and W^k+v overlap, and the volume of the intersection is bigger than 2D_c(W^k+Λ)−1. Returning to the whole spaceR^k, we get the conclusion of the lemma.

Remark8.50. In the case whereΛis the lattice spanned by M_A1,···,A_k, where A₁,· · ·,A_kis a generic family, Lemma8.49can be deduced directly from Lemma8.29(more precisely, improves its conclusion of a factor 2).

9. Of course, this property can be obtained directly by saying that the density is equal to 1 if and only if the rate of injectivity of every matrix of the sequence is equal to 1.

This lemma allows us to give another proof of Theorem8.30(page164), which im-plies in particular that the asymptotic rate of injectivity of a generic sequence of ma-trices of SL_n(R) is smaller than 1/2 (and even is smaller than a sequence converging exponentially fast to 1/2).

Alternative proof of Theorem8.30. Let (A_k)_k≥1 be a bounded sequence of matrices of SL_n(R) andδ>0. As in the previous proof of Theorem8.30, we proceed by induction on kand suppose that the theorem is proved for a rankk∈N^∗. LetΛe_kbe the lattice spanned by the matrixMe_B1,···,B_k (defined by Equation (8.7) page174) and W^k=]−1/2,1/2]^nk be the window corresponding to the model setΓ_k modelled onΛ_k(the lattice spanned by the matrix M_B1,···,B_k, see Equation (8.6)). By Proposition8.40, we have (and n): indeed, for every matrix B∈ SL_n(R), Minkowski theorem implies that there existsx₁∈Zⁿ\ {0}such thatkBx₁k_∞≤1; it then suffices to modify slightly B to decrease What we need is a more precise bound. We apply Corollary8.20to

Λ₁=

Then, the decreasing of the rate of injectivity between timeskandk+ 1 is bigger than the D₁defined in Corollary8.20: using Lemma8.49, we have

D_c From Corollary8.20we deduce that

D(Γ_k+1) = D_c

This proves the theorem for the rankk+ 1.

How to read these figures : The top of the figure represents the set W^k+eΛ_k by the 1-dimensional set [−1/2,1/2] +νZ(in dark blue), for a numberν >1. The bottom of the figure represents the set W^k+1+eΛ_k+1by the set [−1/2,1/2]²+Λ, whereΛis the lattice spanned by the vectors (0,ν) and (1,1−ε) for a parameter ε>0 close to 0. The dark blue cubes represent the “old” cubes, that is, the thickening in dimension 2 of the set W^k+eΛ_k, and the light blue cubes represent the “added” cubes, that is, the rest of the set W^k+1+eΛ_k+1.

Figure 8.17: In the case where the rate is bigger than 1/2, some intersections of cubes appear automatically between timeskandk+ 1.

Figure 8.18: In the case where the rate is smaller than 1/2, there is not necessarily new intersections between timeskandk+ 1.

8.7.3 Proof of Theorem8.24: generically, the asymptotic rate is zero

We now come to the proof of Theorem 8.24. The strategy of proof is identical to that we used in the previous section to state that generically, the asymptotic rate is smaller than 1/2: we will use an induction to decrease the rate step by step. Recall that τ^k(A₁,· · ·,A_k) indicates the density of the set W^k+1+Λ_k(see Definition8.42).

Unfortunately, if the density of W^k+eΛ_k — which is generically equal to the density of thek-th image

bA_k◦ · · · ◦bA₁

(Zⁿ) — is smaller than 1/2, then we can not apply exactly the strategy of proof of the previous section (see Figure8.18). For example, if we take

A₁= 4 1/4

!

and A₂= 1/2 2

! ,

then

cA₂◦cA₁

(Z²) = (2Z)², and for every B₃close to the identity, we haveτ³(A₁,A₂,B₃) = τ²(A₁,A₂) = 1/4.

To overcome this difficulty, we prove that for a generic sequence (Ak)_k≥1, if τ^k(A₁,· · ·,A_k) > 1/`, then τ<sup>k+`−1</sup>(A₁,· · ·,A_k+`−1) is strictly smaller than τ^k(A₁,· · ·,A_k).

An argument of equirepartition (in fact, Propostion8.40) allows to see this problem in terms of area of intersections of cubes. More precisely, we consider the maximal num-ber of disjoint translates of W^k+Λe_k inR^nk: we easily see that if the density of W^k+Λe_k is bigger than 1/`, then there can not be more than` disjoint translates of W^k+Λe_k in R^nk(Lemma8.51). Then, Lemma8.52states that if the sequence of matrices is generic, either the density of W^k+1+eΛ_k+1is smaller than that of W^k+eΛ_k (see Figure8.19), or there can not be more than `−1 disjoint translates of W<sup>k+1</sup>+Λe<sub>k+1</sub>inR<sup>n(k+1)</sup>(see Fig-ure8.20). Applying this reasoning (at most)`−1 times, we obtain that the density of W<sup>k+`−1</sup>+Λe<sub>k+`</sub>−1is smaller than that of W^k+Λe_k. For example if D_c

W^k+Λe_k

>1/3, then

D_c

W^k+2+eΛ_k+2

<D

W^k+Λe_k

(see Figure8.21). To apply this strategy in practice, we have to obtain quantitative estimates about the loss of density we get between timesk andk+`−1.

Remark that with this strategy, we do not need to make “clever” perturbations of the matrices: provided that the coefficients of the matrices are rationally independent, the perturbation of each matrix is made independently from that of the others. However, this reasoning does not tell when exactly the rate of injectivity decreases (likely, in most of cases, the speed of decreasing of the rate of injectivity is much faster than the one obtained by this method), and does not say either where exactly the loss of injectivity occurs in the image sets.

Firstly, we give a more precise the statement of Theorem8.24.

Theorem8.24. For a generic sequence of matrices(A_k)_k≥1ofSL_n(R), for every`∈N, there

In particular, the asymptotic rate of injectivityτ^∞

(A_k)_k≥1

is equal to zero.

The following lemma is a generalization of Lemma8.49. It expresses that if the den-sity of W^k+Λe_k is bigger than 1/`, then there can not be more than` disjoint translates of W^k+Λe_k, and gives an estimation on the size of these intersections.

Lemma 8.51. Let W^k =]−1/2,1/2]^k and Λ⊂ R^k be a lattice with covolume 1 such that of the projection of W^kon the quotient spaceR^k/Λ. As this volume is greater than 1/`, and as the covolume ofΛis 1, the projections of the W<sup>k</sup>+v<sub>i</sub>overlap, and the volume of the points belonging to at least two different sets is bigger than`D_c(W^k+Λ)−1. As there are`(`−1)/2 possibilities of intersection, there existsi,i⁰ such that the volume of the intersection between the projections of W^k+v_iand W^k+v_i⁰is bigger than 2(`D<sub>c</sub>(W<sup>k</sup>+Λ)− 1)/(`(`−1)). Returning to the whole spaceR^k, we get the conclusion of the lemma.

Recall that we denoteΛe_k the lattice spanned by the matrix

Me_A1,···,A_k=

Lemma 8.52. For everyδ>0and everyM>0, there existsε>0and an open set of matrices

Remark8.53. If`= 2, then we have automatically the conclusion (1) of the lemma.

In a certain sense, the conclusion (1) corresponds to the hyperbolic case, and the conclusion (2) expresses that there is a diffusion between timeskandk+ 1.

Proof of Lemma8.52. Let Oε be the set of the matrices B∈ SL_n(R) satisfying: for any collection of vectorsw₁,· · ·, w<sub>`</sub>−1∈R<sup>n</sup>, there exists a set U⊂R<sup>n</sup>/BZ<sup>n</sup>of measure>εsuch that every point of U belongs to at least` different cubes of the collection (Bv+w_i+ W¹)_v∈Zⁿ,1≤i≤`−1. In other words¹⁰, everyx∈Rⁿ whose projectionxonRⁿ/BZⁿbelongs

We easily see that the setsOεare open and that the union of these sets overε>0 is dense (it contains the set of matrices B whose entries are all irrational). Thus, if we are given δ>0 and M>0, there existsε>0 such thatO =Oε isδ-dense in the set of matrices of SL_n(R) whose norm is smaller than M.

We then choose B∈O and a collection of vectors w₁,· · ·, w<sub>`</sub>−1∈R<sup>n</sup>. Letx ∈R<sup>n</sup> be such that x ∈ U. By hypothesis on the matrix B, x satisfies Equation (8.11), so there exists `+ 1 integer vectorsv₁,· · ·, v<sub>`</sub> and ` indicesi₁,· · ·, i_` such that the couples (v_j, i_j) are pairwise distinct and that

∀j∈~1, `+ 1, x∈Bv_j+w_ij+ W¹. (8.12) The following formula makes the link between what happens in then last and in thenpenultimates coordinates ofR^n(k+1):

W^k+1+Λe_k+1+

10. Matrices that does not possess this property are such that the union of cubes form ak-fold tiling.

This was the subject of Furtwängler conjecture, see [Fur36], proved false by G. Hajós. R. Robinson gave a characterization of suchk-fold tilings in some cases, see [Rob79] or [SS94, p. 29].

x

y

Figure 8.19: First case of Lemma8.52, in the case` = 3: the set W^k+1+Λe_k+1 auto-intersects.

x

y

Figure 8.20: Second case of Lemma8.52, in the case ` = 3: two distinct verti-cal translates of W^k+1 + Λe_k+1 intersect (the first translate contains the dark blue thickening of W^k+Λe_k, the second is rep-resented in grey).

Let y be a point belonging to this intersection. Applying Equations (8.12) and (8.14), we get that This implies that the set W^k+1+Λe_k+1auto-intersects (see Figure8.19).

(ii) Or i_j ,i_j⁰ (that is, w_ij ,w_ij0). Combining Equations (8.15) and (8.13) (note that

This implies that two distinct vertical translates of W^k+1+eΛ_k+1intersect (see Fig-ure8.20).

We now look at the global behaviour of all thexsuch thatx∈U. Again, we have two cases.

(1) Either for more than the half of suchx(for Lebesgue measure), we are in the case (i). To each of suchx corresponds a translation vectorw_i. We choosew_i such that the set of corresponding x has the biggest measure; this measure is bigger than

Figure 8.21: Intersection of cubes in the case where the rate is bigger than 1/3. The thickening of the cubes of W^k+Λe_kis represented in dark blue and the thickening of the rest of the cubes of W^k+1+Λe_k+1is represented in light blue; we have also represented another cube of W^k+2+Λe_k+2in yellow. We see that if the projection on thez-axis of the centre of the yellow cube is smaller than 1, then there is automatically an intersection between this cube and one of the blue cubes.

ε/

2(`−1)

≥ε/(2`). Reasoning as in the second proof of Theorem8.30(page177), and in particular applying Corollary8.20, we get that the density D<sub>1</sub> of the auto-intersection of W<sup>k+1</sup>+eΛ<sub>k+1</sub>+ (0, w<sub>i</sub>) is bigger than D<sub>0</sub>ε/(2`). This leads to

D_c(W^k+1+Λe_k+1)<D_c(W^k+Λe_k)−D₀ε 4` . In this case, we get the conclusion (1) of the lemma.

(2) Or for more than the half of such x, we are in the case (ii). Choosing the couple (w_i, w_i⁰) such that the measure of the set of correspondingxis the greatest, we get

D_c

W^k+1+Λe_k+1+ (0^kn, w_i)

∩

W^k+1+eΛ_k+1+ (0^kn, w_i⁰) !

≥ D₀ε (`−1)(`−2). In this case, we get the conclusion (2) of the lemma.

We can now prove Theorem8.24.

Proof of Theorem8.24. As in the previous proofs of such results, we proceed by induc-tion onk. Suppose thatΛe_k is such that D_c(W^k+Λe_k)>1/`. Then, Lemma8.51ensures that it is not possible to have` disjoint translates of W^k+Λe_k. Applying Lemma8.52, we obtain that either D_c(W^k+1+eΛ_k+1)<D_c(W^k+eΛ_k), or it is not possible to have`−1 disjoint translates of W<sup>k+1</sup>+Λe<sub>k+1</sub>. And so on, applying Lemma8.52at most`−1 times, there existsk⁰ ∈~k+ 1, k+`−1such that W^k0+Λe_k⁰ has additional auto-intersections.

Quantitatively, combining Lemmas8.51and8.52, we get D_c

W<sup>k+`−1</sup>+Λe<sub>k+`</sub>−1

≤D

W^k+eΛ_k

− ε 4`

ε

`<sup>2</sup> `−1

2`D_c(W^k+Λe_k)−1

`(`−1) ,

thus

This implies that for every` >0, the sequence of ratesτ<sup>k</sup>is smaller than a sequence con-verging exponentially fast to 1/`: we get Equation (8.9). In particular, the asymptotic rate of injectivity is generically equal to zero.

We now prove the estimation of Equation (8.10). Suppose thatτ^k∈[1/(`−2),1/(`− 1)], we compute how long it takes for the rate to be smaller than 1/(`−1). We apply Equation (8.16)jtimes to`and get

τ<sup>k+j`</sup>−1/`≤λ^j<sub>`</sub>(τ<sup>k</sup>−1/`),

Thus, when`is large enough, the time it takes for the rate to decrease from 1/(`−2) to 1/(`−1) is smaller than`<sup>2(`+1)</sup>=e2(`+1) log`.

. So, when`is large enough, it takes arbitrarily much more time forτ<sup>k</sup>to decrease from 1/(`−2) to 1/(`−1) than forf to decrease from 1/(`−2) to 1/(`−1). By cons,τ(k) =o(f(k)).

In the next part, we will need another technical statement, whose proof is similar to that of Theorem8.24.

Lemma 8.54. There exists an open dense set O of the set`^∞(SL_n(R))of infinite sequences of matrices of SL_n(R) such that for every R₀ > 0, there exists k₀ ∈ N such that for every (A_k)_k≥1∈O, there exists a sequence(w_k)_k≥1of translation vectors belonging to[−1/2,1/2]ⁿ, and a vector ey₀ ∈ Zⁿ, with norm bigger than R₀, such that (where π(A +w) denotes the discretization of the affine mapA +w)

π(A_k0+w_k0)◦ · · · ◦π(A₁+w₁)

(ey₀) =

A_k[₀+w_k0◦ · · · ◦A[₁+w₁

(0) = 0.

Moreover, the pointey₀being fixed, this property can be supposed to remain true on a whole neighbourhood of the sequence(A_k)_k≥1∈O.

Proof of Lemma8.54. In fact, we will prove the following statement.

There exists an open dense setO of the set`^∞(SL_n(R))of infinite sequences of matrices of SL_n(R)such that for everym∈N, there existsk₀∈Nsuch that for every(A_k)_k≥1∈O, there exists a vectory₀∈Zⁿsuch that

Card

dA_k0◦ · · · ◦cA₁⁻1

(y₀)≥m.

The lemma is deduced from this statement by remarking that on the one hand, for every R₀>0, there existsm∈Nsuch that every subset ofZⁿwith cardinality bigger than mcontains at least one point with norm bigger than R₀, and that on the other hand, if we choosew_k∈[−1/2,1/2]ⁿsuch that

w_k= A⁻_k¹

dA_k0◦ · · · ◦A[_k−1

⁻1

(y₀)

modZⁿ,

then the properties of the cardinality of the inverse image ofy₀ are transferred to the point 0.

To prove the statement of the beginning of the proof, we remark that if a loss of injectivity created by the proof of Theorem8.24occurs at a timek, then the intersections of cubes that are created are of the form

W^k+v₁

∩

W^k+v₁+v₂

, with the vectorv₂ having nonzero values only in the 2nlasts coordinates. Thus, if the proof of the theorem createsmintersections of cubes (in times differing of at least 2), there is a point which belongs simultaneously to at leastmcubes. To a point which belongs to this intersection will correspond a pointy₀as desired by the statement. And the perturbation we have made to do that is arbitrarily small, moreover the property remains true on a whole neighbourhood of the perturbed sequence.

The only thing it remains to prove is that the choice of the time k₀is independent from the choice of the sequence of matrices. In fact, we can choosek₀=m². We have two cases. Either in timek₀, the rate is smaller than 1/m, thus automatically the conclusion of the statement will be satisfied by at least one point y₀. Or in time k₀, the rate is bigger than 1/m, thus applying the reasoning abovemtimes we get the conclusion of the statement.

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