Rate of injectivity of linear maps
8.6 Di ff usion process and the case of isometries
v+ (1,0)
•v+ (1,1)
Figure 8.10: The functionϕuin di-mension 2: its value on one vertex of the square is equal to the area of the opposite rectangle; in particu-lar,ϕu(v) is the area of the rectan-gle with the verticesuandv+(1,1).
u0×
(0,0) (1,0)
(0,1) (1,1)
•
•
•
•
Figure 8.11: The red vector is equal to that of Figure8.10foru= Av. If Axbelongs to the bot-tom left rectangle, then π(Ax+ Av) = y ∈ Z2; if Ax belongs to the top left rectangle, then π(Ax+ Av) =y+ (0,1) etc.
8.6 Di ff usion process and the case of isometries
Unfortunately, problems arise when we try to perturb a sequence of matrices to make its asymptotic rate smaller than 1/2. First of all, if the density of an almost peri-odic patternΓ is smaller than 1/2, then the set of differences may not be the full setZn. Even worse, it might happen that this set of differences has big holes, as shown by the following example. We take the almost periodic pattern
Γ = [39
i=0
100Z+i.
For everyx∈Γ andv∈~40,60,x+v<Γ. In other words, for everyv∈~40,60, we have ρΓ(v) = 0, whereas D(Γ) = 0.4. However, the things are not too bad for the frequencies of differences when we are close to 0, as shown by the Minkowski-like theorem for almost periodic patterns (Theorem7.29).
Diffusion process In this paragraph, we study the action of a discretization of a ma-trix on the set of differences of an almost periodic pattern Γ; more precisely, we study the link between the functionsρΓ andρ
bA(Γ).
Foru∈Rn, we define the functionϕu, which is a “weighted projection” ofuonZn. Definition 8.32. Givenu∈Rn, the functionϕu=Zn→[0,1] is defined by
ϕu(v) =
( 0 ifd∞(u, v)≥1
Qn
i=1|1−ui+vi| ifd∞(u, v)<1.
In particular, the function ϕu satisfies P
v∈Znϕu(v) = 1, and is supported by the vertices of the integral unit cube6that contains7u. Figure8.10gives a geometric inter-pretation of this functionϕu.
6. By definition, an integral cube has his faces parallel to the canonical hyperplanes.
7. More precisely, the support ofϕu is the smallest integral unit cube of dimensionn0≤nwhich con-tainsu.
The following property asserts that bA acts “smoothly” on the frequency of differ-ences. In particular, when D(Γ) = D(bAΓ), the functionρ
bAΓ is obtained from the function ρΓ by applying a linear operatorA, acting on each Dirac functionδv in the following way: consider the Dirac functionδAv and spread it on the vertices of the integer cube in which Avbelongs to. More precisely,Aδvis supported by these vertices and its inte-gral is equal to 1, the value assigned to each vertex V is proportional to absolute value of the product of the coordinates of V−Av. Roughly speaking, to compute Aδv, we takeδAv and apply a diffusion process. In the other case where D(bAΓ)<D(Γ), we only have inequalities involving the operatorA to compute the functionρ
bAΓ.
Proposition 8.33. LetΓ ⊂ Zn be an almost periodic pattern and A∈SLn(R) be a generic matrix.
(i) IfD(bA(Γ)) = D(Γ), then for everyu∈Zn, ρbA(Γ)(u) = X
v∈Zn
ϕA(v)(u)ρΓ(v).
(ii) In the general case, for everyu∈Zn, we have D(Γ)
D(bA(Γ))sup
v∈Zn
ϕA(v)(u)ρΓ(v)≤ρ
bA(Γ)(u)≤ D(Γ) D(bA(Γ))
X
v∈Zn
ϕA(v)(u)ρΓ(v).
Remark8.34. This proposition expresses that the action of the discretization of a linear map A on the differences is more or less that of a multivalued map.
Proof of Proposition8.33. We begin by proving the first point of the proposition. We suppose that A∈ SLn(R) is generic and that D(bA(Γ)) = D(Γ). Letx ∈ Γ ∩(Γ −v). We consider the projectionx0of Ax, and the projectionu0of−u=−Av, on the fundamental domain ]−1/2,1/2]n of Rn/Zn. If x0 belongs to the parallelepiped whose vertices are (−1/2,· · ·,−1/2) and u0 (see Figure 8.11), then we obtainπ(A(x+v)) =π(Ax+bAvc) = π(Ax)+bAvc. The same kind of results holds for the other parallelepipeds whose vertices areu0 and one vertex of [−1/2,1/2[n.
The hypothesis of genericity of A ensures that for every v∈Zn, the set Γ ∩(Γ −v), which has density D(Γ)ρΓ(v) (by definition of ρΓ), is equidistributed modulo Zn (by Lemma 7.21). Thus, the points x0 are equidistributed modulo Zn. In particular, the differencev will spread into the differences which are the support of the functionϕAv, and each of them will occur with a frequency given by ϕAv(x)ρΓ(v). The hypothesis about the fact that the density of the sets does not decrease imply that the contributions of each difference ofΓ to the differences ofbA(Γ) add.
In the general case, the contributions to each difference ofΓ may overlap. However, applying the argument of the previous case, we can easily prove the second part of the proposition.
Remark8.35. We also remark that:
(i) the density strictly decreases (that is, D(bA(Γ))<D(Γ)) if and only if there exists v0∈Znsuch thatρΓ(v0)>0 andkAv0k∞<1;
(ii) if there existsv0∈Znsuch that X
v∈Zn
ϕA(v)(v0)ρΓ(v0)>1, then the density strictly decreases by at leastP
v∈ZnϕA(v)(v0)ρΓ(v0)−1;
(iii) we can compute which differences will go to the difference 0, that is, the dif-ferences u ∈ Zn such that there exists x ∈ Γ ∩(Γ −u) such that bA(x) = bA(x+u) (that will make the rate of injectivity decrease). The set of such differences u is A−1(B(0,1))∩Zn (recall that B(0,1) is an infinite ball). By Hajós thorem (The-orem 8.21, see also Corollary 8.22), this set contains a non zero vector when A is generic. More generally, the set of differences u ∈ Γ ∩(Γ −u) such that bA(x) +v = bA(x+u) is A−1(B(v,1))∩Zn. Iterating this process, it is possible to compute which differences will go to 0 in timet.
Rate of injectivity of a generic sequence of isometries Proposition 8.33 allows to prove an alternative version of Theorem8.24for isometries.
Theorem 8.36. Let(Pk)k≥1be a generic sequence of matrices ofOn(R). Thenτ∞((Pk)k) = 0.
Figure 8.12: Images of Z2by discretizations of rotations, a point is black if it belongs to the image ofZ2 by the discretization of the rotation. From left to right and top to bottom, anglesπ/4,π/5,π/6,π/7,π/8 andπ/9.
In the previour section, the starting point of the proof was Lemma 8.29, which ensures that when the rate of injectivity is bigger than 1/2, the frequency ofany dif-ference is bigger than a constant depending on the rate. Here, the starting point is the Minkowski theorem for almost periodic patterns (Theorem7.29), which givesone nonzero difference whose frequency is positive. The rest of the proof of this theorem consists in using again an argument of equidistribution. More precisely, we apply suc-cessively the following lemma, which asserts that given an almost periodic pattern Γ of density D0, a sequence of isometries andδ>0, then, perturbing each isometry of at mostδif necessary, we can make the density of thek0-th image ofΓ smaller thanλ0D0, withk0andλ0depending only on D0andδ. The proof of this lemma involves the study of the action of the discretizations on differences made in Proposition8.33
Lemma 8.37. Let(Pk)k≥1be a sequence of matrices of On(R), δ>0andΓ ⊂Zn an almost periodic pattern. Then there existsk0=k0(D(Γ))(decreasing in D(Γ)),λ0=λ0(D(Γ),δ)<1
Figure 8.13: Successive images ofZ2 by discretizations of random rotations, a point is black if it belongs to (dRθk◦ · · · ◦dRθ1)(Z2), where theθiare chosen uniformly randomly in [0,2π]. From left to right and top to bottom,k= 1,2,3,5,10,50.
(decreasing inD(Γ)and inδ), and a sequence(Qk)k≥1of totally irrational matrices ofOn(R), such thatkPk−Qkk ≤δfor everyk≥1and
D
(dQk0◦ · · · ◦Qc0)(Γ)
<λ0D(Γ).
We begin by proving that this lemma implies Theorem8.36.
Proof of Theorem8.36. Suppose that Lemma8.37is true. Let τ0 ∈]0,1[ and δ>0. We want to prove that we can perturb the sequence (Pk)kinto a sequence (Qk)kof isometries, which isδ-close to (Pk)kand is such that its asymptotic rate is smaller thanτ0(and that this remains true on a whole neighbourhood of these matrices).
Thus, we can suppose that τ∞((Pk))>τ0. We apply Lemma 8.37to obtain the pa-rametersk0=k0(τ0/2) (becausek0(D) is decreasing in D) and λ0=λ0(τ0/2,δ) (because λ0(D,δ) is decreasing in D). Applying the lemma ` times, this gives a sequence (Qk)k of isometries, which is δ-close to (Pk)k, such that, as long as τ`k0(Q0,· · ·,Q`k0)> τ0/2, we haveτ`k0(Q1,· · ·,Q`k0)<λ`0D(Zn). But for` large enough,λ`0<τ0, which proves the theorem.
Proof of Lemma8.37. The idea of the proof is the following. Firstly, we apply the Minkowski-type theorem for almost periodic patterns (Theorem 7.29) to find a uni-form constant C>0 and a pointu0∈ Zn\ {0} whose norm is “not too big”, such that ρΓ(u0)> CD(Γ). Then, we apply Proposition8.33 to prove that the differenceu0 inΓ eventually goes to 0; that is, that there exists k0 ∈N∗ and an almost periodic pattern eΓ of positive density (that we can compute) such that there exists a sequence (Qk)k of isometries, withkQi−Pik ≤δ, such that for everyx∈eΓ,
(dQk0◦ · · · ◦Qc1)(x) = (dQk0◦ · · · ◦Qc1)(x+u0).
Figure 8.14: Expectation of the rate of injectivity of a random sequences of rotations:
the graphic represents the mean of the rate of injectivityτk(Rθk,· · ·,Rθ1) depending on k, 1≤k ≤200, for 50 random draws of sequences of angles (θi)i, with eachθi chosen independently and uniformly in [0,2π]. Note that the behaviour is not exponential.
We begin by applying the Minkowsky-like theorem for almost periodic patterns (Theorem7.29) to aeuclidean ball B0R with radius R (recall that [B] denotes the set of integer points inside B):
1 Card[B0R]
X
u∈[B0R]
ρΓ(u)≥D(Γ)Card[B0R/2]
Card[B0R] . (8.2)
An easy estimation of the number of integer points inside of the balls B0Rand B0R/2leads to8:
Card[B0R/2] Card[B0R] ≥
√
R/2−√ n/2
√ R +√
n/2
!2n
,
in particular, if R≥n, we obtain Card[B0R/2]
Card[B0R] ≥
√ 2−1
3
!2n
≥ 1 53n, thus Equation (8.2) becomes
1 Card[B0R]
X
u∈[B0R]
ρΓ(u)≥D(Γ) 1
53n. (8.3)
We now want thatρΓ(0) = 1 plays only a little role in Equation (8.3), that is 1
Card[B0R]≤ D(Γ) 2×53n,
8. See for example [Fri82, page 5], this book also performs a complete investigation on the subject of finding the number of integer points in a euclidean ball.
using the same estimate as previously for Card[B0R], that is true if a direct calculation shows that it is true for example if
R = R0=n+ 212 pn
D(Γ). (8.4)
In this case, Equation (8.3) implies 1
We now take R≥R0, and perturb each matrix Pk into a totally irrational matrix Qk such that for every pointx∈[B0R]\ {0}, the point Qk(x) is far away from the latticeZn. More precisely, as the set of matrices Q∈ On(R) such that Q([B0R])∩Zn,{0}is finite, there exists a constantd0(R,δ) such that for every P∈On(R), there exists Q∈On(R) such thatkP−Qk ≤δand for everyx∈[B0R]\ {0}, we haved∞(Q(x),Zn)> d0(R,δ). Applying Lemma 7.21 (which states that if the sequence (Qk)k is generic, then the matrices Qk are “non resonant”), we build a sequence (Qk)k≥1of totally irrational matrices of On(R) such that for everyk∈N∗, we have:
– kPk−Qkk ≤δ;
– for everyx∈[B0R]\ {0}, we haved∞(Qk(x),Zn)> d0(R,δ);
– the set Qk◦Q[k−1◦ · · · ◦Qc1(Γ) is equidistributed moduloZn.
We then consider the difference u0(given by Equation (8.5)). We denote bybPc(u) the point of the smallest integer cube of dimensionn0≤nthat contains P(u) which has the smallest euclidean norm (that is, the point of the support ofϕP(u)with the smallest euclidean norm). In particular, if P(u)<Zn, thenkbPc(u)k2<kP(u)k2(wherek · k2is the euclidean norm). Then, the point (ii) of Proposition8.33shows that
ρQc1(Γ)(bQ1c(u0))≥ D(Γ)
We then remark that the euclidean norm ofbQ1c(u0) can only take a finite number of values (it lies in
√
Z). Then, there existsk0≤R2such that bQk0c ◦ · · · ◦ bQ1c
(u0) = 0;
in particular, by Equation (8.4), we have
k0≤2n2+ 89 888 (D(Γ))n/2.
Then, point (ii) of Remark8.35implies that the density of the image set satisfies
D
Remark8.38. We could be tempted to try to apply this strategy of proof to any sequence of matrices in SLn(R). However, this does not work in the general case, because if we consider a non-euclidean norm N, there could be somex∈Rnsuch that for every vertex yof the cube containingx, we have N(y)≥N(x).
We now set out a conjecture which states a generic dichotomy for the behaviour of the frequency of differences for discretizations of generic sequences of matrices, in the same vein as [Boc02] (for dimension 2) or [ACW14] (in the general case).
Conjecture 8.39. Let(Ak)k≥1be a generic sequence of matrices ofSLn(R). Then, if we note Γk = (bAk◦ · · · ◦bA1)(Zn), the standard deviation of ρΓk is either very small, either as big as possible. More precisely, the standard deviation of
1 q
D(Γk)
1−D(Γk)ρΓk
either tends to 0 askgoes to infinity (which corresponds to the case of zero Lyapunov expo-nent), or tends to 1 askgoes to infinity (which corresponds to the hyperbolic case).
The idea underlying the conjecture is that for for every ε>0, for every generic se-quence (Ak)k≥1of matrices and everykbig enough, we have two cases. corresponds to the zero Lyapunov exponent case, and the idea is that diffusion process wins on the hyperbolicity of the sequence of matrices.
– Or corresponds to the hyperbolic case and the idea is that the hyperbolicity of the sequence of matrices wins on the diffusion process.