Study of the continuity of the rate of injectivity

In this section, we study the properties of continuity of the functionτ. We first show that there exists some matrices in whichτis not continuous (Proposition8.1). However, we show in Proposition8.4thatτis continuous at every totally irrational matrix; more precisely, the mapτcoincides with themean rate of injectivityτ(Definition8.3) on the set of totally irrational matrices, and the mapτis continuous on the whole set GL_n(R).

We finish this section in estimating the lack of continuity ofτat the matrices which are not totally irrational (Proposition8.7).

In the following example, we show that on some rational matrices, the rate of injec-tivity is not continuous. In particular, it will give us an example where Lemma7.16is not true when there is no restriction about the value ofv on I_Q(A) (the set of indices corresponding to rational coefficients, see Notation7.13).

Our counterexample is given by irrational perturbations of the matrix f₀=

It can be easily seen that the rate of injectivity of f₀ is 1/2 (see Figure8.1). Remark that it depends on the choice made for the projection ofR²onZ²: if we make the same choice for both directions (that is what we have chosen in Definition7.11) then the rate of injectivity is 1/2, otherwise this rate of injectivity is 1. Forε“small”, we consider the linear map

Proposition 8.1. The rate of injectivity off_εtends ³₄ whenε<Qtends to 0.

In particular, the rate of injectivityτis not continuous inf₀.

Lemma 8.2. – For allv∈]0,¹₂[²∪]¹₂,1[²moduloZⁿ, the rate of injectivity off₀+vis ¹₂. – For allv∈]0,¹₂[×]¹₂,1[∪]¹₂,1[×]0,¹₂[moduloZⁿ, the rate of injectivity off₀+vis1.

Proof of Lemma8.2. We want to know when it is possible to find two different vectors x, y ∈Zⁿ such that π(f₀x+v) =π(f₀y+v). This implies thatd∞(f₀x, f₀y)<1, a simple calculation (see Figure8.1) shows that in this case,x=y±(1,0). Asf₀(2Z×Z)⊂Z², we can suppose thatx= 0 ; we want to know if fory=±(1,0), we haveπ(f₀x+v) =π(f₀y+v), that isπ(v) =π(v±(1/2,1/2)). Again, a simple calculation shows that this occurs if and only ifv∈]0,¹₂[²∪]−¹

2,0[².

Thus, for half of the vectorsv(for Lebesgue measure), the rate of injectivity off₀+v is 1 and for the other half of the vectorsv, the rate of injectivity off₀+v is ¹₂. Propo-sition8.1 then follows from an argument of equirepartition. To make it rigorous, we define the mean rate of injectivity.

Definition 8.3. For A∈GL_n(R), the quantity τ(A) =

Z

Tⁿ

τ(A +v) dv

is called themean rate of injectivityof A.

We say that a matrix A∈GL_n(R) istotally irrational if the image A(Zⁿ) is equidis-tributed³moduloZⁿ; in particular, this is true when the coefficients of A form aQ-free family.

The motivation of this definition is that the mean rate of injectivity is continuous and coincides with the rate of injectivity on totally irrational matrices.

Proposition 8.4. The mean rate of injectivityτis continuous onGL_n(R). Moreover, ifAis totally irrational, thenτ(A) =τ(A); and even more,τ(A) =τ(A +v)for everyv∈Tⁿ.

Thus, the restriction of the rate of injectivity to totally irrational matrices is a con-tinuous function, which extends to GL_n(R) into a continuous function. In particular, the restriction of this function to any bounded subset of SL_n(R) is uniformly continuous.

Proposition 8.4, combined with Lemma 8.2, obviously implies Proposition 8.1 (as forε<Q, the mapf_εis totally irrational).

Proof of Proposition8.4. The continuity of τ will be obtained later as a direct conse-quence of Corollary8.13.

Let A∈GL_n(R) be a totally irrational matrix. We want to prove thatτ(A) =τ(A). To do that, we show thatτ(A) =τ(A+v) for everyv∈Tⁿ. As{Ax|x∈Zⁿ}is equidistributed moduloZⁿ, for everyv∈Tⁿ, there existsx∈Zⁿsuch thatd∞(v−Ax,Zⁿ)<ε. But as the density is a limit independent from the choice of the centre of the ball (Corollary7.7 and Theorem7.12), we have

Rlim→+∞

Card

π(A(Zⁿ))∩B_R

Card[B_R] = lim

R→+∞

Card

π(A(Zⁿ) + Ax)∩B(Ax,R)

Card[B_R] .

This proves thatτ(A) =τ(A + Ax). We then use Remark7.18, which states that as A is totally irrational, the functionv7→τ(A+v) is continuous. As the vector Axis arbitrarily close to the vectorv, we getτ(A) =τ(A +v).

3. It is equivalent to require that it is dense instead of equidistributed; it is also equivalent to ask that Zⁿis equidistributed modulo A(Zⁿ).

Remark 8.5. The same proof shows that if Γ ⊂ Zⁿ is an almost periodic pattern, and A∈GL_n(R) is such that A(Γ) is equidistributed moduloZⁿ (which is true for a generic A, see Lemma7.21), then B7→D

bB(Γ)

is continuous in B = A.

We now study the behaviour of τ at the matrices where it is not continuous. In particular, we prove that the set of matrices on which the rate of injectivity makes “big jumps” is small. More precisely, we define the oscillation of a map.

Definition 8.6. For A∈GL_n(R), theoscillationofτin A is the quantity ω_τ(A) = lim

Proposition 8.7. For everyε>0, the setn

A∈GL_n(R)|ω_τ(A)>εo

is locally contained in a finite union of hyperplanes.

To prove this proposition, we will need a technical lemma which requires the fol-lowing definition.

Definition 8.8. LetΛbe a lattice ofRⁿ. Then [Bou98, Chap. VII, §1, 2] implies that the Z-moduleΛ+Zⁿcan be decomposed into two complementary modules

Λ+Zⁿ=Λ_continuous⊕Λ_discrete,

such thatΛ_continuousis dense in the vector space it spans andΛ_discreteis discrete.

During the proof of Proposition8.7, we will make use of the following lemma, that we will not prove.

Lemma 8.9. For every ε > 0, the set of matrices A ∈ GL_n(R) such that (AZⁿ)_discr has a fundamental domain of diameter smaller than εcontains the complement of a locally finite union of hyperplanes.

Proof of Proposition8.7. Letε>0. We consider a matrix A∈GL_n(R) such that (AZⁿ)_discr has a fundamental domainD of diameter smaller thanε; Lemma8.9asserts that such matrices contains the complement of a locally finite union of hyperplanes. We use Proposition8.4, which states thatτ is continuous and coincides withτ on totally irra-tional matrices, to estimate the oscillation ofτ in A : it is smaller than the maximal value of

D(π(Λ+v))−D(π(Λ+v⁰))

, when the vectorsv andv⁰ run throughTⁿ. First of all, we remark that the map

v7→D(π(Λ+v)) isΛ+Zⁿ-periodic. Thus, we only have to estimate

D(π(Λ+v))−D(π(Λ+v⁰))

forvandv⁰ inD. We denote by (Zⁿ)⁰ the set of points inRⁿwhose discretization is not canonically defined, more precisely,

The proposition then follows from the fact that this last quantity is small (uniformly in ε).

Chapter 8. Rate of injectivity of linear maps 155

Figure 8.2: First geometric construction: the green points are the elements of Λ, the blue parallelogram is a fundamental domain ofΛand the grey squares are centred on the points ofΛand have radii 1/2; their union form the set U. A pointx∈Zⁿbelongs toπ(Λ) if and only if it belongs to at least one grey square.