A geometric viewpoint on the mean rate

In this section, we present two geometric constructions to compute the rate of injec-tivity of a matrix.

First construction Let A∈ GL_n(R) and Λ = A(Zⁿ). The density of π(Λ) is the pro-portion ofx∈Zⁿ belonging toπ(Λ); in other words the proportion ofx∈Zⁿ such that there existsλ∈Λwhose distance tox(fork · k_∞) is smaller than 1/2. This property only depends on the value ofxmoduloΛ. If we consider the union

U =[

then we obtain the following formula (using also Equation (7.7), which linksτ(A) and D(Λ)).

In particular, when the matrix A is totally irrational, the measureν is the uniform measure; thus ifD is a fundamental domain ofRⁿ/Λ, thenτ(A) is the area ofD ∩U.

The same holds for the mean rate of injectivity and any matrix (not necessarily totally irrational).

Proposition 8.11. For everyA∈GL_n(R), τ(A) =|det(A)|Leb

pr_Rn/Λ(U)

=|det(A)|Leb(U∩D).

Thus, the mean rate of injectivity can be seen as the area of an intersection of parallelepipeds (see Figure8.2).

Figure 8.3: Mean rate of injectivity on SL₂(R) on the fundamental domain D (more precisely, D ={z ∈C| |z|>1,Imz >0,−1/2<Rez <1/2}) of the modular surface, for various angles in T¹D (0,π/10,π/5, 3π/10, 2π/5,π/2).

With the same kind of arguments, we easily obtain a formula forρ

bA(Zⁿ)(v) (the fre-quency of the differencev inbA(Zⁿ), see Definition7.24).

Proposition 8.12. IfA∈GL_n(R)is totally irrational, then for everyv∈Zⁿ, ρbA(Zⁿ)(v) = Leb

B(v,1/2)∩U .

Sketch of proof of Proposition8.12. We want to know which proportion of pointsx ∈ Γ are such that x+v also belongs to Γ. But modulo Λ = A(Zⁿ), x belongs to Γ if and only ifx∈B(0,1/2). Similarly,x+v belongs toΓ if and only if x∈B(−v,1/2). Thus, by equirepartition,ρ

bA(Zⁿ)(v) is equal to the area of B(v,1/2)∩U.

From Proposition8.11, we deduce the continuity ofτ.

Corollary 8.13. The mean rate of injectivity is a continuous function onGL_n(R), which is locally polynomial with degree smaller thannin the coefficients of the matrix.

Moreover, this construction gives a quick algorithm to compute numerically the mean rate of injectivity of some matrices: this algorithm is much more efficient than the naive one consisting in computing the cardinality of the image of a large ball by the discretization, see Figure8.3and Figure8.4.

It also allows to compute simply the mean rate of injectivity of some examples of matrices.

Application8.14. Forθ∈[0,π/2], the mean rate of injectivity of a rotation ofR²of angle θis (see Figure8.5).

τ(R_θ) = 1−(cos(θ) + sin(θ)−1)².

Figure 8.4: Action of the geodesic flow on the mean rate of injectivity: the red corre-sponds to a rate of injectivity close to 1 and the blue to a rate of injectivity close to 0.

The figure represents the rate of injectivity on geodesics of the space SL₂(R)/SL₂(Z), identified with the unitary tangent space T¹S of the modular surface S starting from the pointi, depending on the time (on the vertical axis) and on the starting angle of the geodesic (on the horizontal axis). The mean rate of injectivity of each lattice has been computed in using Proposition8.18

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Figure 8.5: Computation of the mean rate of injectivity of a rotation ofR²: it is equal to 1 minus the area of the interior of the red square.

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Figure 8.6: Computation of the mean rate of injectivity of a lattice having one vector parallel with the horizontal axis with length` <1.

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Figure 8.7: Computation of the mean rate of injectivity of a lattice having one vector parallel with the horizontal axis with length` ∈]1,2[ for a parameter x >

`−1.

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Figure 8.8: Computation of the mean rate of injectivity of a lattice having one vector parallel with the horizontal axis with length` ∈]1,2[ for a parameterx <

`−1.

Application8.15. We consider a lattice ofR²of covolume 1 with one basis vector parallel with the horizontal axis. We have several cases:

1. If the length` of the vector parallel with the horizontal axis is smaller than 1, then the mean rate of injectivity is equal to`, independently from the choice of the second basis vector (see Figure8.6).

2. If the length`is bigger than 1 but smaller than 2, we choosev= (x,1/`) another basis vector of the lattice. We can suppose that−`/2≤x≤`/2; by symmetry we only treat the casex∈[0, `/2]. The mean rate of injectivity is then a piecewise affine map with respect tox:

– if`−1≤x≤`/2, then the mean rate of injectivity is constant and equal to 1−(`− 1)(2/`−1) =`−2 + 2/`(see Figure8.7);

– if 0≤x≤`−1, then the mean rate of injectivity is equal to 1−(`−1)(2/`−1)−(1− 1/`)(`−1−x) = 1/`+x−x/`(see Figure8.8).

We do not treat the other cases where` >2.

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Figure 8.9: Second geometric construction: the green points are the elements ofΛ, the blue square is a fundamental domain of Zⁿ and the grey squares are centred on the points ofΛand have radii 1/2. The rate of injectivity is equal to the integral on the blue square of the inverse of the number of different squares the point belongs to.

Second construction Here, we take another viewpoint to compute the rate of injec-tivity of A. To begin with, we remark that for every mapf defined on a finite set E, the

cardinality off(E) can be computed by the formula Thus, we are reduced to compute Card

bA⁻¹({A(x)ˆ })

whereψis the function (different from the function1_Uused previously; recall thatΛ= AZⁿ) ψ=X

λ∈Λ

1_B(λ,1/2)

which is the sum of the indicator functions of the balls of radius1/2centred on the points of Λ(see Figure8.9).

Proof of Lemma8.16. We denote by ˜x the projection of Axon the fundamental domain B_1/2 of Zⁿ. We want to prove that there exists k different vectors y ∈ Zⁿ such that

then, by combining Equation (8.1) with Lemma8.16, we obtain Proposition 8.17.

And we have a similar statement for the mean rate of injectivity.

Proposition 8.18.

These properties can be also directly deduced from the first geometric construction by applying a double counting argument. We state it in more general context because we will need a more precise statement in the sequel (here, we only need the casem= 0, thus B₁ = B); this lemma allows to compute the area of the projection of a set on a quotient by a lattice.

Lemma 8.19. LetΛ₁be a subgroup ofR^mwhich is a lattice in the vector space it spans,Λ₂

In particular, the area of B₂(the projection on the quotient by the direct sum of both lattices) is smaller than (or equal to) that of B₁(the projection on the quotient by the first lattice). The loss of area is given by the following corollary.

Corollary 8.20. With the same notations as for Lemma8.19, if we denote by D₁= Lebn Proof of Lemma8.19. Consider the function

φ: Rⁿ/Λ₁ −→N^∗

Obviously, the rate of injectivityτ(A) of any matrix of A∈SL_n(Z) is equal to 1. In Section8.1, we have found other examples of affine maps with determinant 1 whose rate of injectivity is also 1. Also, in Application8.15, taking `= 1, this gives another class of examples where the rate of injectivity is one. In this section, we investigate more in detail this question: what are the matrices with determinant 1 whose mean rate of injectivity is 1? With Proposition 8.11, we can reformulate this question in terms of intersection of cubes: if det(A) = 1, τ(A) = 1 if and only if the cubes B(λ,1/2), with λ∈AZⁿ, tile the spaceRⁿ. This is a classical problem raised by H. Minkowski in 1896 (see [Min10]), and answered by G. Hajós in 1941.

Theorem 8.21 (Hajós, [Haj41]). LetΛ be a lattice of Rⁿ. Then the collection of squares {B(λ,1/2)}_λ∈Λ tiles the plane if and only if in a canonical basis of Rⁿ (that is, permuting coordinates if necessary),Λadmits a generating matrix which is upper triangular with ones on the diagonal.