Hausdorff dimension of affine random covering sets in torus

www.imstat.org/aihp 2014, Vol. 50, No. 4, 1371–1384

DOI:10.1214/13-AIHP556

© Association des Publications de l’Institut Henri Poincaré, 2014

Hausdorff dimension of affine random covering sets in torus

Esa Järvenpää

, Maarit Järvenpää

, Henna Koivusalo

, Bing Li

and Ville Suomala

aDepartment of Mathematical Sciences, P.O. Box 3000, 90014 University of Oulu, Finland.

E-mail:esa.jarvenpaa@oulu.fi;maarit.jarvenpaa@oulu.fi;henna.koivusalo@oulu.fi;ville.suomala@oulu.fi

bDepartment of Mathematics, South China University of Technology, Guangzhou, 510641, P.R. China. E-mail:libing0826@gmail.com Received 2 July 2012; revised 20 February 2013; accepted 27 February 2013

Abstract. We calculate the almost sure Hausdorff dimension of the random covering set lim sup_n→∞(g_n+ξ_n)ind-dimensional torusT^d, where the setsg_n⊂T^dare parallelepipeds, or more generally, linear images of a set with nonempty interior, andξ_n∈T^d are independent and uniformly distributed random points. The dimension formula, derived from the singular values of the linear mappings, holds provided that the sequences of the singular values are decreasing.

Résumé. Nous calculons presque sûrement la dimension de Hausdorff de l’ensemble de recouvrement aléatoire lim sup_n→∞(gn+ ξ_n)dans le toreT^dde dimensiond, oùg_n⊂T^d sont des parallélépipèdes, ou plus généralement, des images linéaires d’un en- semble d’intérieur non vide etξn∈T^dsont des points aléatoires indépendants et uniformément distribués. La formule de dimen- sion, exprimée en fonction des valeurs singulières des applications linéaires, est valable à condition que la suite de ces valeurs singulières soit décroissante.

MSC:60D05; 28A80

Keywords:Random covering set; Hausdorff dimension; Affine Cantor set

1. Introduction

Given a sequence of positive numbers(l_n)and a sequence of independent random variables(ξ_n)uniformly distributed on the circleT¹=R/Z, definethe random covering setEas follows:

E=

x∈T¹|x∈ [ξ_n, ξ_n+l_n]for infinitely manyn

=lim sup

n→∞ [ξ_n, ξ_n+l_n].

Denoting the Lebesgue measure by L and using the Borel–Cantelli lemma and Fubini’s theorem, it follows that, almost surely, the following dichotomy holds:

L(E)=

0, when_∞

n=1l_n<∞, 1, when_∞

n=1l_n= ∞, (1.1)

that is, almost all or almost no points of the circle are covered, depending on whether or not the series of the lengths of the covering intervals diverges.

The case of full Lebesgue measure has been extensively studied. It was a long-standing problem to find conditions on(ln)guaranteeing that the whole circle is covered almost surely, that is,

P

E=T¹

=1. (1.2)

This problem, known in literature as the Dvoretzky covering problem, was first posed by Dvoretzky [5] in 1956. After substantial contribution of many, including Kahane [18], Erd˝os [7], Billard [3] and Mandelbrot [24], the full answer was given by Shepp [29] in 1972. He proved that (1.2) holds if and only if

∞ n=1

1

n²exp(l1+ · · · +l_n)= ∞,

where the lengths(ln)are in decreasing order. After this, a natural problem, raised by Carleson (private communication to Kahane), is to describe the growth of the covering number of a given pointx∈T¹, that is, to study the asymptotic behaviour of the sums

C_N(x)= N

n=1

χ_[ξn,ξn+ln](x),

whereχAis the characteristic function of a setA. Obviously, the expectationE(CN(x))=N

n=1ln. In the caseln=^γ_n withγ >1, Fan and Kahane [10] proved that almost surely the order of the covering numberCN(x)is logNfor every x∈T¹, meaning that for sufficiently largeN

A_γlogN≤min

x∈T¹C_N(x)≤max

x∈T¹C_N(x)≤B_γlogN

with positive and finite constantsA_γ andB_γ. Furthermore, Fan [9] verified that the set Fβ=

x∈T¹ lim

N→∞

C_N(x) N

n=1ln

=β

has positive Hausdorff dimension for a certain interval ofβ >0 in the casel_n=^γ_n withγ >0. For generall_n, Barral and Fan [2] answered Carleson’s problem by identifying three kinds of phenomena depending whether the index

¯

γ =lim sup_N→∞

N n=1ln

−logl_N is zero, positive and finite or infinite. More precisely, when γ¯ =0, dimHF_β =1 almost surely for allβ≥0, whenγ¯= ∞,F1=T¹almost surely, and when 0<γ <¯ ∞, dimHFβ depends onβ. Here the Hausdorff dimension is denoted by dimH.

For the case of zero Lebesgue measure, the Hausdorff dimension ofEwas first calculated by Fan and Wu [12]

in the caseln=1/n^α. When studying the Hausdorff measure and the large intersection properties ofEfor general ln, Durand [4] gave another, independent proof of the dimension result. According to [12] and [4], the almost sure Hausdorff dimension ofEis given by

dimHE=inf

t≥0 ^∞

n=1

l_n^t <∞

=lim sup

n→∞

logn

−logln

, (1.3)

where the lengthsln are in decreasing order. In [4], the author also proved that the packing dimension ofEequals 1 almost surely. When considering the hitting probability property of the random setE, Li, Shieh and Xiao [22]

provided an alternative way to obtain the Hausdorff and packing dimension results under some additional conditions.

The result (1.3) can be also proven as a consequence of the mass transference principle due to Beresnevich and Velani [1] (see Proposition4.7). The fact that both packing and box counting dimensions are equal to 1 almost surely follows sinceEis almost surely a denseGδ-set inT¹(see [19], Chapter 5, Proposition 11, and [27], Section 2).

In this paper we study random covering sets ind-dimensional torusT^d. Letting(gn)be a sequence of subsets of T^d and letting(ξn)be a sequence of independent random variables, uniformly distributed onT^d, define the random covering set by

E=lim sup

n→∞ (gn+ξn)= ∞ n=1

∞ k=n

(gk+ξk).

Notice that a counterpart of (1.1) is easily obtained, that is, almost surely L(E)=

0, when_∞

n=1L(g_n) <∞, 1, when_∞

n=1L(g_n)= ∞, whereLis the Lebesgue measure onT^d.

On thed-dimensional torus the Dvoretzky covering problem has been studied by El Hélou [6] and Kahane [20]

among others. In [20] Kahane gave a complete solution for the problem when the setsg_n are similar simplexes (see also Janson [16]). However, in the general case the covering problem has not been completely solved.

For an overview on the research on random covering sets and related topics, we refer to [19], Chapter 11, the survey [21] and the references therein. Here we only mention a few variations on the classical random covering model. For example, Hawkes [13] considered under which conditions all the points inK⊂T¹are covered with probability one (or zero). Mandelbrot [25], in turn, introduced Poisson covering of the real line (see also Shepp [28]). In general metric spaces, the random coverings by balls have been studied by Hoffman-Jörgensen [15]. Recent contributions to the topic include various types of dynamical models, see Fan, Schmeling and Troubetzkoy [11], Jonasson and Steif [17] and Liao and Seuret [23].

We address the question of determining the analogue of (1.3) in higher dimensional case. In [12] the method is strongly adapted to the 1-dimensional case whereas the argument based on the mass transference principle [1] can be carried through in any dimension provided that the setsg_nare uniformly ball like (see Proposition4.7). Our main interest is the case where the setsg_nare not uniformly ball like, and therefore, the mass transference principle cannot be applied. It turns out that almost surely the Hausdorff dimension of the covering set E is given in terms of the singular value functions of the linear mappings related to the system, see Theorem2.1.

To this end, in Section2we introduce our setting, state our main result and prove preliminary lemmas including the upper bound for the dimension. In Section3 we construct a random subset of the covering setE having large dimension with positive probability which, in turn, gives the lower bound of the dimension in Section4.

2. Preliminaries and statement of main theorem

Denote the closed ball of radiusrand centrexinR^dbyB(x, r). LettingL:R^d→R^dbe a contractive linear injection, the imageL(B(0,1))of the unit ballB(0,1)is an ellipse whose semiaxes are non degenerated. The singular values 0< αd(L)≤ · · · ≤α1(L) <1 ofLare the lengths of the semiaxes ofL(B(0,1))in decreasing order. Given 0< s≤d, define thesingular value functionby

Φ^s(L)=α1(L)· · ·αm−1(L)αm(L)^s−(m−1), wheremis the integer such thatm−1< s≤m.

We use the notationsT^dfor thed-dimensional torus andLfor the Lebesgue measure onT^d. Consider a probability space(Ω,A, P )and let(ξ_n)be a sequence of independent random variables which are uniformly distributed onT^d, that is,(ξ_n)_∗P =L, where(ξ_n)_∗P is the image measure ofP underξ_n. Letting(g_n)be a sequence of subsets ofT^d, we use the notationG_n for the random translatesG_n=g_n+ξ_n⊂T^d and define the random covering set generated by(gn)by

E=E^ω=lim sup

n→∞ G_n.

In this paper we consider the casegn=Π (Ln(R)), whereR⊂ [0,1]^d has non-empty interior,Ln:R^d→R^d is a contractive linear injection for alln∈NandΠ:R^d→T^dis the natural covering map. Moreover, we assume that for alli=1, . . . , dthe sequence of singular valuesα_i(L_n)decreases to 0 asntends to infinity. Defining

s₀=inf

0< s≤d ^∞

n=1

Φ^s(L_n) <∞

, (2.1)

with the interpretations0=d if_∞

n=1Φ^d(Ln)= ∞, we are ready to state our main theorem.

Theorem 2.1. ForP-almost allω∈Ωwe have

dimHE^ω=s₀. (2.2)

Theorem2.1is an immediate consequence of the following proposition concerning the case where each generating setgn is a rectangular parallelepiped inT^d meaning that there exist a parallelepipedg˜n⊂R^d such thatgn=Π (g˜n).

In what follows rectangular parallelepipeds will consistently be called rectangles.

LetE(g_n)=E^ω(g_n)be the covering set generated by a sequence(g_n)of rectangles. For all rectanglesgand for all 0< s≤d define

Φ^s(g)=α₁(g)· · ·α_m−1(g)α_m(g)^s−(m−1),

where 0< αd(g)≤ · · · ≤α1(g) <1 are the lengths the edges ofgin decreasing order andmis the integer such that m−1< s≤m.

Proposition 2.2. Assume that(g_n)is a sequence of rectangles such that for alli=1, . . . , d the sequence of lengths α_i(g_n)decreases to0asntends to infinity.Then almost surely

dimHE(g_n)=s₀(g_n), (2.3)

where

s0(gn)=inf

0< s≤d ^∞

n=1

Φ^s(gn) <∞

with the interpretations₀=dif_∞

n=1Φ^d(g_n)= ∞.

We proceed by verifying first that Theorem2.1follows from Proposition2.2.

Proof of Theorem2.1as a consequence of Proposition2.2. Letting(L_n),RandEbe as in Theorem2.1, there are sequences(g_n)and(gn)of rectangles such thatg_n⊂Π (Ln(R))⊂gn, and moreover,αi(g_n)=cαi(Ln)andαi(gn)= cαi(Ln)for alli=1, . . . , d. Here the constantsc andcare independent ofnandi. SinceE(g_n)⊂E⊂E(gn)we have

dimHE g_n

≤dimHE≤dimHE(gn).

Applying Proposition2.2to the sequences(g_n)and(g_n)and noting thats₀(g_n)=s₀(g_n)=s₀, gives (2.2).

It remains to prove Proposition2.2. As the first step we verify the following lemma according to which the Haus- dorff dimension ofE(g_n)is always bounded above bys₀(g_n). The proof is standard following, for example, the ideas in [8].

Lemma 2.3. Assume that(gn)ands0(gn)are as in Proposition2.2.Then for allω∈Ω we havedimHE^ω(gn)≤ s₀(g_n).

Proof. We may assume thats₀(g_n) < d. Lets₀(g_n) < s < d and letmbe the integer withm−1< s≤m. For each n∈Nwe estimate the number of cubes of side lengthα_m(g_n)needed to coverG_n. By expanding the lastd−m+1 edges ofGnto lengthαm(gn)and by dividing the expanded rectangle to cubes of side lengthαm(gn), we end up with an upper bound

α₁(g_n) α_m(g_n)

+1

. . .

α_m−1(g_n) α_m(g_n)

+1

≤2^m−1α₁(g_n)· · ·α_m−1(g_n)α_m(g_n)^−m+1, where the integer part of anyx≥0 is denoted byx.

Recalling that for allN∈N E(g_n)⊂ ^∞

n=N

G_n,

gives the following estimate for thes-dimensional Hausdorff measure H^s(E)≤lim inf

N→∞

∞ n=N

2^m−1√

dα_m(g_n)s

α₁(g_n)· · ·α_m−1(g_n)α_m(g_n)^−m+1

=lim inf

N→∞

∞ n=N

2^m−1(√

d)^sΦ^s(gn)=0.

This implies that dimHE(g_n)≤s₀(g_n).

We continue by proving two auxiliary results.

Lemma 2.4. Assume that(L_n)is a sequence of contractive linear injectionsL_n:R^d→R^d.Lets₀be as in(2.1)and letm−1< s₀≤m.Defining for allm−1< s < s₀

f (s):=lim sup

n→∞

logn

−logΦ^s(Ln), we havef (s) >1.

Proof. We will show thatf (s)≥1 for allm−1< s < s0andf is strictly decreasing. This clearly implies the claim.

Letm−1< s < s₀. The fact that_∞

n=1Φ^s(L_n)= ∞implies that for allε >0 there exists a subsequence(n_k) such thatΦ^s(Ln_k) > ¹

n¹_k^+ε for allk. From this we deduce thatf (s)≥₁₊¹_ε, and lettingεgo to 0 yieldsf (s)≥1.

Considerδ >0 such thatm−1< s+δ < s₀. SinceΦ^s(L_n)≥α_m(L_n)^s we obtain Φ^s+δ(Ln)=Φ^s(Ln)αm(Ln)^δ≤Φ^s(Ln)^1+δ/s,

giving

f (s+δ)≤lim sup

n→∞

logn

(1+δ/s)(−logΦ^s(Ln))= f (s)

1+δ/s < f (s).

Hencef is strictly decreasing.

Remark 2.5. Lemma2.4holds for all0< s < s0,but this stronger claim is not necessary for our purposes.

Proposition 2.6. Assume that G⊂T^d andL(G) >0.Letξ₁, . . . , ξ_n be independent,uniformly distributed random variables onT^d.Let

M_n=#

i∈ {1, . . . , n}|ξ_i∈G ,

where#Adenotes the number of elements in a setA.Then P

M_n≤1

2nL(G)

≤4(1−L(G)) nL(G) .

Proof. Denote byχ_Athe characteristic function of a setA. Calculating the first and second moments ofM_ngives E(M_n)=E

_n

i=1

χ_{ξi∈G}

=nL(G)

and E

M_n²

=E _n

i=1

χ_{ξ_i∈G}

2

=E _n

i=1

χ_{ξ_i∈G}+

j=i

χ_{ξ_i∈G}χ_{ξ_j∈G}

=nL(G)+ n²−n

L(G)². From Chebyshev’s inequality we deduce

P

M_n≤1 2E(M_n)

≤P M_n−E(M_n) ≥1 2E(M_n)

≤4(E(M_n²)−E(Mn)²)

E(Mn)² =4(1−L(G)) nL(G)

which completes the proof.

3. Construction of random Cantor sets

Let(g_n)ands₀(g_n)be as in Proposition2.2. Consider an integermsuch thatm−1< s₀(g_n)≤m. For notational sim- plicity, we assume that 0 is a vertex of eachg_n. Indeed, by choosing suitable deterministic translates, we find an iso- morphic probability space(Ω,A, P )where this is the case since the random variables(ξn)are uniformly distributed and the rectangles(g_n)are deterministic. For eachn, letT_n:R^d→R^dbe a linear map such thatΠ (T_n([0,1]^d))=g_n. Observe thatα_i(T_n)=α_i(g_n)for alli=1, . . . , d. Letm−1< s < s₀(g_n). For the purpose of proving Proposition2.2 we construct in this section an eventΩ(∞)⊂Ω, having positive probability, and a random Cantor like setC^ω such thatC^ω⊂E^ωfor allω∈Ω(∞). In Section4we prove that dimHC^ω≥salmost surely conditioned onΩ(∞).

Leta0= ¹₂. Consider a sequence(al)of real numbers larger than 1/2 increasing to 1 with _∞

l=1 1

a_l <∞. By Lemma2.4, there exists a sequence(n_k)of natural numbers satisfying

klim→∞

logn_k

−logΦ^s(Tn_k)=f (s) >1. (3.1)

Moreover, by considering a suitable subsequence of(nk), we may assume that for allk∈N diam(gnk)≤1

2(1−a_k−1)α_d(g_nk−1), (3.2)

n_kL(g_nk−1)≥n⁽³_k^{+f (s))/(2+2f (s))} and (3.3)

lognk≥nk−1, (3.4)

wheren0=0 andg0=T^d. Notice that since the sequence(nk)is deterministic it is independent ofω∈Ω.

We proceed by constructing inductively a random nested sequence of finite collectionsCkof rectangles as follows:

LetC0= {T^d}andN₀=1. DefineN₁= ¹₂a^d₀n₁andI (1,T^d)= {1, . . . , N1}. For alli∈I (1,T^d), letg_i be a linear isometric copy ofg_n1 contained ing_i. The existence ofg_i follows from the fact thatα_j(g_n1)≤α_j(g_i)for alli≤n₁ andj=1, . . . , d. For eachi∈I (1,T^d), setG_i=g_i+ξi. ThenG_i⊂Gi. DefiningC1= {G_i|i∈I (1,T^d)}, we have

G∈C1

G⊂

n1

i=1

G_i.

Furthermore, the collectionC1can be chosen for anyω∈Ω=:Ω(1)givingP (Ω(1))=q₁withq₁=1.

Assume that there exist events Ω(1), . . . , Ω(k−1) with P (_k−1

j=1Ω(j )) =q₁· · ·q_k−1 such that for all ω∈ _k−1

j=1Ω(j )there are collectionsC1, . . . ,Ck−1having the following properties for allj=1, . . . , k−1 1

2 3

a_j^d₋₁(n_j−n_j−1)L(g_nj−1)≤N_j≤(n_j−n_j−1)L(g_nj−1), whereN_j=#Cj, (3.5)

G∈Cj

G⊂

G∈Cj−1

G, (3.6)

#

G ∈Cj|G ⊂G

= 1

2a^d_j−1m_jL(g_nj−1)

for eachG∈Cj−1

wherem_j=

n_j−n_j−1 N_j−1

, (3.7)

Cj is a finite collection of isometric copies ofg_nj and (3.8)

G∈Cj

G⊂

nj

l=n_j−1+1

G_l. (3.9)

We define an event Ω(k)such that P (_k

j=1Ω(j ))=q₁· · ·q_k and for all ω∈_k

j=1Ω(j )there is a collectionCk

satisfying (3.5)–(3.9). WriteCk−1= {G₁, . . . ,G_Nk−1}and setm_k= ^nk_N⁻_kⁿ₋^k−1₁ . Forl=1, . . . , Nk−1, define random sets

I (k,G_l)= i∈

n_k−1+1+(l−1)mk, . . . , n_k−1+lm_k

|ξ_i∈a_k−1G_l ,

whereaGis the similar copy ofGwith similarity ratioaand with the same centre asG. Let Ω(k)=

ω∈Ω #I (k, G) > 1

2a_k^d₋₁m_kL(g_nk−1)for allG∈Ck−1

and

q_k=P

Ω(k)|Ω(1), . . . , Ω(k−1) .

Note thatqk>0. For eachG∈Ck−1we denote byI (k, G)the collection of the first¹₂a^d_k−1mkL(gn_k−1)elements in I (k, G)and set

Ck=

G_i|G∈Ck−1, i∈I (k, G)

and N_k=#Ck,

whereG_i=g_i+ξi andg_i is a linear isometric copy ofgn_k contained ingi. (See Fig.1.) Observe thatNk is deter- ministic. As above,g_i exists sinceα_j(g_nk)≤α_j(g_i)for allj=1, . . . , d andi≤n_k. Clearly, (3.7) and (3.8) are valid forCk. Since, by inequality (3.2), we haveg_i+ξ_i⊂G∈Ck−1provided thatξ_i ∈a_k−1G, property (3.6) holds forCk. Furthermore, the choices ofmkandI (k, Gl)imply (3.9). The choice ofmkgives

1 2

3

a_k^d₋₁(n_k−n_k−1)L(g_nk−1)≤N_k−1 1

2a_k^d₋₁m_kL(g_nk−1)

=N_k

≤(n_k−n_k−1)L(g_nk−1), and therefore, condition (3.5) is satisfied forCk. Finally,

P _k

l=1

Ω(l)

=P

Ω(k)|Ω(1), . . . , Ω(k−1) P

_k−1

l=1

Ω(l)

=q1· · ·qk.

Fig. 1. Construction ofCk.

LettingΩ(∞)=_∞

n=1Ω(n), we haveP (Ω(∞))=_∞

n=1q_n. Define for allω∈Ω(∞) C^ω=^∞

n=1

G∈Cn

G⊂E^ω.

Next we verify that the Cantor like setC^ω ⊂E^ω exists with positive probability. We use the notationFk for the σ-algebra generated by the random variablesξ₁, . . . , ξ_nk.

Proposition 3.1. With the above notation we haveP (Ω(∞)) >0.

Proof. We have q_k=P

Ω(k)|Ω(1), . . . , Ω(k−1)

= 1

P (k−1 l=1Ω(l))P

Ω(k)∩

k−1

l=1

Ω(l)

= 1

P (k−1

l=1Ω(l))E

E(χΩ(k)χk−1

l=1Ω(l)|Fk−1)

= 1

P (_k−1

l=1Ω(l))E χk−1

l=1Ω(l)E(χ_Ω(k)|Fk−1)

= 1 P (_k−1

l=1Ω(l))E χk−1

l=1Ω(l)E(χ

G∈Ck−1{#I (k,G)>(1/2)a_k−1^d mkL(g_nk−1)}|Fk−1)

≥ 1

P (_k−1

l=1Ω(l))E

χk−1 l=1Ω(l)

1−

G∈Ck−1

E(χ_{{#I (k,G)≤(1/2)m}

kL(a_k−1g_nk−1)} Fk−1)

,

and applying Proposition2.6hence gives q_k≥1−N_k²₋₁ 8(1−L(a_k−1g_nk−1))

(n_k−n_k−1)L(a_k−1g_nk−1)=:1−p_k.

Inequalities (3.5) and (3.4), in turn, imply thatN_k−1≤(n_k−1−n_k−2)L(g_nk−2)≤n_k−1≤logn_k, and therefore, noting thatn_k−n_k−1≥¹₂n_kby (3.4) and using (3.3), we obtain

∞ k=1

p_k≤ ∞ k=1

8(1−L(a_k−1g_nk−1))(logn_k)² (1/2)a_k^d₋₁n_kL(g_nk−1) ≤

∞ k=1

8(logn_k)²

(1/2)a_k^d₋₁n⁽³_k^{+f (s))/(2+2f (s))}

<∞,

where the convergence follows since by (3.4) the sequence(n_k)is growing exponentially fast. Lettingk₀∈Nbe such thatp_k<1 for allk≥k₀, we have_∞

k=1q_k≥k0

k=1q_k∞

k=k0+1(1−p_k) >0.

Remark 3.2. The idea of finding a large-dimensional Cantor subset of the random covering set was already exploited in the dimension calculation of Fan and Wu[12]in the case ofT¹.In their proof it is essential that the setsC^ω are homogeneous and the construction intervals are well-separated,which follows from well-known results on random spacings of uniform random samples[14].Structure of the set allows them then to directly estimate sizes of inter- sections of balls with the set C^ω,giving the dimension bound from below.In our choice of the subsetC^ω,however, separation of the generating sets plays no role.Indeed,it is a well-known fact that for self-affine sets no separation condition guarantees the dimension formula.Also a direct estimate for measures of balls is probably hopeless.Instead a potential theoretic method based on a transversality argument is the key,see Lemma4.3below.In the implementation of this idea we need the assumption(3.8).

4. Dimension estimate

Using the notation introduced in Section3, we prove that fors < s₀(g_n)the event{ω∈Ω(∞)|dimHC^ω≥s}has pos- itive probability. To obtain the dimension bound, we use potential theoretic methods and define a measure supported onC^ωwith finites-energy. In what follows, we consider only the eventΩ(∞)and denote the expectation overΩ(∞) simply byE.

For any ω∈Ω(∞),k∈N andG∈Ck−1, let M_k=#I (k, G)= ¹₂a^d_k−1m_kL(g_nk−1)be the number of levelk construction rectangles contained in G. Notice that M_k is a deterministic number depending only on k. For later notational simplicity, we will relabel the random variablesξ_i using a deterministic tree structure.

For alll∈N, consider the setsJl= {i₁. . . i_l|i_j∈ {1, . . . , Mj}for allj∈ {1, . . . , l}}and defineJ=_∞

l=0Jl, with the conventionJ0= {∅}. Fori,j∈J, denote byi∧jthe maximal common initial sequence ofiandjand letij∈Jbe the word obtained by juxtaposing the wordsiandj. Further, we denote by|i|the length ofi∈J, that is,|i| =lifi∈Jl. For eachl≤kandi∈Jl, define thecylinderof lengthland of depthkbyC(i, k)= {j∈Jk|i∧j=i}. Fori∈ {1, . . . , M1}, defineφ_i=ξ_i andG(i)=g_i+φ_i and letT_i be a linear map such that Π (T_i([0,1]^d)=g_i. Assume that we have defined the random variables φ_i and the rectangles G(i)∈Ck−1 for all i∈Jk−1. Let I (k, G(i))= {j₁, . . . , j_Mk} wherej_i < j_i+1 in the natural order given by the construction. For alli∈ {1, . . . , Mk}, defineφ_ii =ξ_ji,g_ii =g_j

i

and G(ii)=g_ii +φ_ii and let T_ii be a linear map satisfying Π (T_ii([0,1]^d))=g_ii. Then det(Tn_|i|)=L(G(i)) and Φ^s(T_i)=Φ^s(Tn_|i|)for alli∈J. For notational purposes setG(∅)=T^dandφ_∅=0. When necessary we viewT_i as a map onT^dby identifyingT^dwith[0,1[^d. Finally, fori1, . . . ,ik∈J, denote byF(i1, . . . ,ik)theσ-algebra generated by the events{ω∈Ω(∞)|G(il)=Ql for alll=1, . . . , k}, where eachQl⊂T^d is an isometric copy ofgn_|il|. Remark 4.1. Note that {φ_i|i ∈ C(j, k)} = {ξ_i|i ∈ I (k, G(j))} for any j ∈ Jk−1 and {φ_i|i ∈ Jk} = {ξ_i|i ∈

G∈Ck−1I (k, G)}.Let A⊂T^d be a Borel set withL(A) >0.Since ξj is uniformly distributed onT^d for givenj,

everyξ_jis uniformly distributed onAwhen conditioned on the eventξ_j∈A.Leti∈Nand letii∈Jk+1.By definition φ_ii=ξ_j for somej∈ {n_k+1, . . . , nk+1}withξ_j∈G(i),and hence the random variableφ_ii is uniformly distributed ona_kG(i)when conditioned onφ_ii=ξ_j and theσ-algebraF(i).Furthermore,

E

χ_{φii∈A}|F(i)

=

n_k+1

j=n_k+1

E

χ_{φii∈A}|F(i), φii=ξ_j E

χ_{φii=ξj}|F(i)

=L(A∩a_kG(i)) L(a_kG(i))

n_k+1

j=nk+1

E

χ_{φii=ξj}|F(i)

=L(A∩a_kG(i)) L(a_kG(i)) .

Henceφ_ii is uniformly distributed insidea_kG(i)when conditioned onF(i).Moreover,ifjsatisfiesj∧ii=ii,con- ditioning onF(i,j)instead ofF(i)does not change the uniform distribution of φii onakG(i),sinceξj and ξl are independent forj=l.Recall that even though the corner pointsφ_ii andφ_ihare independent fori=h,the rectangles G(ii)andG(ih)are not,since the orientation ofg_ii is determined by the indexji.

Lemma 4.2. The sequence of measuresμ^ω_l onT^dgiven by

μ^ω_l =

i∈Jl(T_i +φ_i)_∗L

N_l (4.1)

converges in weak^∗-topology to a measureμ^ω supported onC^ω.

Proof. By the Riesz representation theorem a weak^∗-limitμ^ω exists, if we prove that for all positive, continuous functionsf onT^dthe sequence

fdμ^ω_l converges.

To that end, fix a positive, continuous functionf onT^dandε >0. SinceT^d is compact, there existsδ >0 with

|f (x)−f (y)|< εfor all|x−y|< δ. LetKbe so large that diam(gnK) < δ, and fixk≥K. Writeμ^ω_k as a sum of measuresμ^ω_i,kdefined by

μ^ω_k =

i∈J_K

j∈C(i,k)

(T_j+φj)_∗L

Nk =

i∈J_K

μ^ω_i,k.

For alli∈JK, we haveμ^ω_i,k(G(i))=_N¹_K =μ^ω_i,K(G(i))and sptμ^ω_i,k⊂G(i). Therefore,

fdμ^ω_k −

fdμ^ω_K ≤

i∈JK

G(i)

fdμ^ω_i,k−

G(i)

fdμ^ω_i,K ≤ε, since diamG(i)=diam(gnK) < δ. Thus sequence

fdμ^ω_l converges. The claim sptμ^ω ⊂C^ω holds since C^ω is compact and sptμ^ω_l ⊂

G∈ClGfor alll.

Next we show that for alls < s₀(g_n)thes-energyI^s(μ^ω)= ^dμω_|^(x)dμ_x−y|s^ω(y) ofμ^ω is finite almost surely. In the energy estimate we will make use of the following lemma [8], Lemma 2.2.

Lemma 4.3 (Falconer). Letsbe non-integral with0< s < d and letT:R^d→R^dbe an affine injection.Then there exists a number0< D₀<∞,depending only ondands,such that

[0,1]^d

dL(x)

|T (x)|^s ≤ D0

Φ^s(T ).

Lemma 4.4. For alli,j∈Jandx, y∈T^d we have

E

χG(j)(y)χG(i)(x)

≤ _∞

l=1

1 a_l

2d

det(Tn_|i|)det(Tn_|j|) det(Tn_|i∧j|)² E

χG(i∧j)(y)χG(i∧j)(x) .