AN UPPER-BOUND FOR THE HAUSDORFF DIMENSION OF LIMSUP SETS

Abstract

In this article, we establish an upper-bound theorem for the Haus-dor dimension of limsup sets. This theorem together with a theorem of extraction of sub-sequences of balls are used to prove the sharpness of certain lower-bound estimates established via mass transference principles.

Full text

AN UPPER-BOUND FOR THE HAUSDORFF DIMENSION OF
LIMSUP SETS
EDOUARD DAVIAUD
UNIVERSITÉ PARIS-EST, LAMA (UMR 8050), UPEMLV, UPEC, CNRS, F-94010,
CRÉTEIL, FRANCE
Abstract.
In this article, we establish an upper-bound theorem for the Haus-
dor dimension of limsup sets. This theorem together with a theorem of ex-
traction of sub-sequences of balls are used to prove the sharpness of certain
lower-bound estimates established via mass transference principles.
Contents
1. Introduction 1
2. Denitions and recalls 2
2.1. Hausdor dimension of sets and measures 3
2.2. The
µ
-a.c. covering property and an extraction result 4
3. Main statements 5
3.1. An upper-bound theorem for the dimension of limsup sets 5
3.2. Study of the optimality of mass transference principles for self-similar
measures 7
4. Proof of Theorem 2.2 8
4.1. Extraction of weakly redundant
µ
-a.c subsequences 9
4.2. Extraction of sub-sequences of balls with conditioned measure 12
5. Proof of Theorem 3.1 17
5.1. Proof of item
(1)
of Theorem 3.1 17
5.2. Proof of item
(2)
of Theorem 3.1 20
5.3. A toy example: detecting sequences of too large balls 21
References 23
1.
Introduction
Investigating Hausdor dimensions of sets of points approximable at a certain
rate by a given sequence of points
(xn)n∈N
is a classical topic in Diophantine
approximation (see [5] and [4] among other references), in dynamical systems [15,
21, 23, 12] and in multifractal analysis [19, 2, 3, 22]. These studies consist in
general, knowing that
µ(lim supn→+∞Bn:= B(xn, rn)) = 1
for a certain measure
µ
and a sequence of radii
(rn)n∈N
, in investigating the Hausdor dimension of
lim supn→+∞Un
where
Un⊂Bn
. Typically
Un
is a contracted ball
Bδ
n:= B(xn, rδ
n)
with same center as
Bn
, but recently, general open sets
Un
have been considered
[20, 24, 6, 10]. In such situations, the so-called mass transference principles are
designed to provide a lower bound for the Hausdor dimension (or the Hausdor
measure) of
lim supn→+∞Un
.
1

2 E. DAVIAUD
For instance, when
µ
is the Lebesgue measure and
(Bn)n∈N
is a sequence of balls
of
[0,1]d
, Beresnevich and Velani established in [5] that if
µ(lim supn→+∞Bn) = 1
,
then for every
δ > 1
and any ball
B
Hd
δ(B∩lim sup
n→+∞
Bδ
n) = +∞,
where
Hd
δ
denotes the
d
δ
-dimensional Hausdor measure. This result extends in
particular the following result previously established by Jaard ([18])
dimH(lim sup
n→+∞
Bδ
n)≥d
δ.
In order to obtain a general lower-bound when the measure involved is not the
Lebesgue measure or an Alfhors regular one but any measure
µ∈ M(Rd)
and the
sets
(Un)n∈N
are not shrunk balls but are only assumed to be open, the
µ
-essential
content of a set (see Denition 3.1) was introduced in [9].
These works raise the natural question of whether one can obtain an upper-
bound theorem for
dimHlim supn→+∞Un
which involves geometric quantities that
are similar to the essential content. The main theorem of this article, Theorem
3.1, establishes such a result.
The lower-bounds provided by mass transference principles are empirically suited
for situations where the balls of the sequence
(Bn)n∈N
of comparable radii do not
intersect too much (for instance it works well for dyadic cubes, rational balls etc...).
In particular Theorem 3.1 below largely relies on taking into account account the
possible gain of dimension of
lim supn→+∞Un
due to potential overlaps between
the balls
(Bn)n∈N
at same scale. For a precise statement see Theorem 3.1.
This result together with a technical (but useful) extraction theorem (Theorem
2.2) allows to conclude that the mass transference principles for self-similar mea-
sures established in [9, Theorem 2.11] and in [9, Theorem 2.13] are optimal in a
satisfying sense and that the bound for the mass transference from ball to rectan-
gle in the case of the Lebesgue measure established in [25]. It also hows that the
mass transference principle from ball to rectangle in the case of a quasi-Bernoulli
measure on the dyadic grid established in [6] are optimal as well.
2.
Definitions and recalls
Let
d∈N
.
For
n∈N
, the set of dyadic cubes of generation
n
of
Rd
is denoted
Dn(Rd)
and
dened as
Dn(Rd) = ¶Qd
i=1[ki
2n,ki+1
2n)©(k1,...,kd)∈Zd.
For
x∈Rd
,
r > 0
,
B(x, r)
stands for the closed ball of (
Rd
,
|| ||∞
) of center
x
and radius
r
. Given a ball
B
,
|B|
is the diameter of
B
.
For
t≥0
,
δ∈R
and
B:= B(x, r)
,
tB
stand for
B(x, tr)
, i.e. the ball with
same center as
B
and radius multiplied by
t
, and the
δ
-contracted
Bδ
is dened
by
Bδ=B(x, rδ)
.
Given a set
E⊂Rd
,
◦
E
stands for the interior of the
E
,
E
its closure and
∂E
is
the boundary of
E
, i.e,
∂E =E\◦
E.

AN UPPER-BOUND FOR THE HAUSDORFF DIMENSION OF LIMSUP SETS 3
The
σ
-algebra of Borel sets of
Rd
is denoted by
B(Rd)
,
Ld
is the Lebesgue
measure on
B(Rd)
and
M(Rd)
stands for the set of Borel probability measure over
Rd.
For
µ∈ M(Rd)
,
supp(µ) := x∈Rd:∀r > 0, µ(B(x, r)) >0
is the topolog-
ical support of
µ.
Given
E∈ B(Rd)
,
dimH(E)
and
dimP(E)
denote respectively the Hausdor and
the packing dimension of
E
.
2.1.
Hausdor dimension of sets and measures.
Denition 2.1.
Let
ζ:R+7→ R+
be an increasing mapping verifying
ζ(0) = 0
.
The Hausdor measure at scale
t∈(0,+∞)
associated with
ζ
of a set
E
is dened
by
(1)
Hζ
t(E) = inf (X
n∈N
ζ(|Bn|) : (Bn)n∈N
closed balls,
|Bn| ≤ t
and
E⊂[
n∈N
Bn).
The Hausdor measure associated with
ζ
of a set
E
is dened by
(2)
Hζ(E) = lim
t→0+Hζ
t(E).
For
t∈(0,+∞)
,
s≥0
and
ζ:x→xs
, one simply uses the usual notation
Hζ
t(E) = Hs
t(E)
and
Hζ(E) = Hs(E).
In particular, the
s
-dimensional Hausdor
outer measure at scale
t∈(0,+∞]
of the set
E
is dened by
(3)
Hs
t(E) = inf (X
n∈N
|Bn|s: (Bn)n∈N
closed balls,
|Bn| ≤ t
and
E⊂[
n∈N
Bn).
For
s≥0,
the outer-measure
Hs
∞
(obtained for
t= +∞
) is referred as the
s
-
dimensional Hausdor content.
Denition 2.2.
Let
µ∈ M(Rd)
. For
x∈supp(µ)
, the lower and upper local
dimensions of
µ
at
x
are
dim(µ, x) = lim inf
r→0+
log µ(B(x, r))
log r
and
dim(µ, x) = lim sup
r→0+
log µ(B(x, r))
log r.
Then, the lower and upper dimensions of
µ
are dened by
(4)
dimH(µ) =
infess
µ(dim(µ, x))
and
dimP(µ) =
supess
µ(dim(µ, x)).
It is known that (for more details see [11])
dimH(µ) = inf
E∈B(Rd): µ(E)>0dimH(E)
and
dimP(µ) = inf
E∈B(Rd): µ(E)=1 dimP(E).
A measure verifying
dimH(µ) = dimP(µ) := α
will be called an
α
-exact di-
mensional measure. From Denition 2.2, such measures verify, for
µ
-almost every
x∈Rd
,
limr→0+log µ(B(x,r))
log r=α.
Alfhors-regular measures (so in particular the Lebesgue measure) are for instance
exact-dimensional.

4 E. DAVIAUD
2.2.
The
µ
-a.c. covering property and an extraction result.
In this section,
we recall some denitions stated in [8] and the extraction theorem mentioned in
introduction is stated.
2.2.1.
The
µ
-a.c. covering property.
The notion of
µ
-asymptotically covering se-
quences of balls was introduced in order to highlight a key covering property used
in the proof of the KGB-Lemma [5]. The denition is the following.
Denition 2.3
([8])
.
Let
µ∈ M(Rd)
. The sequence of balls
B= (Bn)n∈N
of
Rd
is
said to be
µ
-asymptotically covering (in short,
µ
-a.c) when there exists a constant
C > 0
such that for every open set
Ω⊂Rd
and
g∈N
, there is an integer
NΩ∈N
as well as
g≤n1≤... ≤nNΩ
such that:
• ∀ 1≤i≤NΩ
,
Bni⊂Ω,
• ∀ 1≤i=j≤NΩ
,
Bni∩Bnj=∅,
•
one has
(5)
µ [
1≤i≤NΩ
Bni!≥Cµ(Ω).
The set
{Bni}1≤i≤NΩ
is called a
(C, g, µ)
-covering of
Ω.
The following result justies the introduction of Denition 2.3 when one studies
limsup sets of balls.
Theorem 2.1
([8])
.
Let
µ∈ M(Rd)
and
B= (Bn)n∈N
be a sequence of balls of
Rd
with
limn→+∞|Bn|= 0
.
(1) If
B
is
µ
-a.c, then
µ(lim supn→+∞Bn)=1.
(2) If there exists
v < 1
such that
µlim supn→+∞(vBn)= 1
, then
B
is
µ
-a.c.
Let us also mention that it is known that item
(1)
is an equivalence as soon as
the measure is doubling (see the proof of the KGB-lemma in [5]).
2.2.2.
An extraction theorem suited to
µ
-a.c. sequences of balls.
In this sub-section
we state an extraction theorem of sub-sequences of balls which preserves the
µ
-a.c.
property.
Let us start by recalling the following notion, introduced in [2].
Denition 2.4
([2])
.
Let
B= (Bn=: B(xn, rn))n∈N
be a family of balls in
Rd
.
Denote by
Tk(B) = Bn: 2−k−1< rn≤2−k.
The family
B
is said to be weakly
redundant when for all
k
, there exists an integer
Jk
and
Tk,1(B), .., Tk,Jk(B)
a par-
tition of
Tk(B)
such that:
(C1)Tk(B) = S1≤j≤JkTk,j (B),
(C2)
for every
1≤j≤Jk
and every pair of balls
B=B′∈ Tk,j (B)
,
B∩B′=∅,
(C3) limk→+∞log2(Jk)
k= 0.
We refer to the second subsection of Section 4 for more details about the weak
redundancy property. In particular Proposition 4.4 highlights the link between
overlapping sequences of balls and weakly redundant sequences.
Our extraction theorem is the following.
Theorem 2.2.
Let
µ∈ M(Rd)
Let
(Bn)n∈N
be a sequence of balls of
Rd.

AN UPPER-BOUND FOR THE HAUSDORFF DIMENSION OF LIMSUP SETS 5
(1) If
(Bn)n∈N
is
µ
-a.c, then there exists a
µ
-a.c sub-sequence
(Bϕ(n))n∈N
which
is weakly redundant.
(2) If there exists
v < 1
such that
µ(lim supn→+∞vBn) = 1
, then there exists
a
µ
-a.c sub-sequence
(Bϕ(n))n∈N
verifying
(6)
dimH(µ)≤lim inf
n→+∞
log µ(Bϕ(n))
log |Bϕ(n)|≤lim sup
n→+∞
log µ(Bϕ(n))
log |Bϕ(n)|≤dimP(µ).
Remark 2.3.

Theorem 2.2 implies in particular that if the sequence of
balls
(Bn)n∈N
veries
µ(lim supn→+∞vBn)=1
, for some
v < 1
, it is possi-
ble to extract a
µ
-a.c sub-sequence verifying items
(1)
and
(2)
.

The left-hand side of
(6)
actually only requires that the sequence
(Bn)n∈N
satises
µ(lim supn→+∞Bn)=1
or
(Bn)n∈N
is
µ
-a.c. (and the sequence
(Bϕ(n))n∈N
satises then the same hypothesis).
In the next section, one states the main theorem of this article.
3.
Main statements
3.1.
An upper-bound theorem for the dimension of limsup sets.
3.1.1.
Statement of the upper-bound theorem.
In [9], the
mu
-essential content of
a set was introduced in order to establish mass transference principles in settings
where the ambient measure is not Alfhors-regular.
Denition 3.1
([9])
.
Let
µ∈ M(Rd)
, and
s≥0
. The
s
-dimensional
µ
-essential
Hausdor content at scale
t∈(0,+∞]
of a set
A⊂ B(Rd)
is dened as
(7)
Hµ,s
t(A) = inf {Hs
t(E) : E⊂A, µ(E) = µ(A)}.
Our main theorem is the following.
Theorem 3.1.
Let
(Bn)n∈N
a sequence of balls of
Rd
such that
|Bn| → 0
and
(Un)n∈N
a sequence of open sets satisfying for every
n∈N, Un⊂Bn
. For
n∈N,
set
(8)
e
Un=[
p∈N:Bp⊂3Bn
and
1
2≤|Bn|
|Bp|≤2
Up.
Let
0≤s≤d.
(1)
If there exists
µ∈ M(Rd)
such that
Hs
∞(e
Un∩lim supp→+∞Bp)≤µ(3Bn)
for all
n∈N
, then
(9)
dimH(lim sup
n→+∞
Un)≤s.
(2)
Conversely, if there exists
µ∈ M(Rd)
with
dimH(µ)≥s
such that, for all
n∈N,Hµ,s
∞(e
Un)≥µ(3Bn)
and
µ(lim supp→+∞Bp) = 1
then
dimH(lim sup
n→+∞
Un)≥s.
(3)
Assume furthermore that
(Bn)n∈N
is weakly redundant. Then items
(1)
and
(2)
hold with
Un
instead of
e
Un
,
Bn
instead of
3Bn
and in item
(2),
the
assumption that
(Bn)n∈N
is
µ
-a.c. instead of
µ(lim supp→+∞Bp) = 1.

6 E. DAVIAUD
Item
(2)
will be obtained as a consequence of the mass transference established
in [9] and stated as Theorem 5.9 in this paper so the new result is the upper-bound
provided by item
(1)
.
As mentioned in the introduction, the lower-bounds provided by mass trans-
ference principles are usually suited for sequences of sets or balls which, in an
appropriate sense (see Denition 2.4), do not overlap too much. When this is not
the case, one needs to take into account the gain of dimension due to potential
accumulation of many sets
Un
. In Theorem 3.1, the sets
e
Un
are introduced for this
purpose.
Note that, in the settings of item
(2)
, since
µ(lim supp→+∞Bp)=1,
one has
Hs
∞(e
Un∩lim sup
p→+∞
Bp)≥ Hµ,s
∞(e
Un)≥µ(3Bn).
In this regard, item
(1)
can be seen as a partial counterpart of item
(2)
.
Remark 3.2.
The constant
3
in Theorem 3.1 and
2
in
(8)
is meant so that
e
Un⊂3Bn.
These constants have little importance for practical applications. In fact one
could replace both of them by any
v > 1
. Moreover, the constant
3
can be dropped
as soon as the measure is doubling.
3.1.2.
Application to self-similar settings.
For the sake of simplicity, the results
below are stated for self-similar measures but completely rely on estimates of the
Hausdor essential content ([9, Theorem 2.6]). Since similar estimates holds for
weakly conformal measures ([7]), it is not hard to see that these result also holds
for weakly conformal measures (see [14] for a denition of a weakly conformal IFS).
Denition 3.2.
A self-similar IFS is a family
S={fi}m
i=1
of
m≥2
contracting
similarities of
Rd
.
Let
(pi)i=1,...,m ∈(0,1)m
be a positive probability vector, i.e.
p1+· · · +pm=
1
. The self-similar measure
µ
associated with
{fi}m
i=1
and
(pi)m
i=1
is the unique
probability measure such that
(10)
µ=
m
X
i=1
piµ◦f−1
i.
The topological support of
µ
is the attractor of
S
, that is the unique non-empty
compact set
K⊂X
such that
K=Sm
i=1 fi(K)
.
The existence and uniqueness of
K
and
µ
are standard results [17]. Recall that
due to a result by Feng and Hu [14] any self-similar measure is exact dimensional.
Given
S
a self-similar IFS and
K
its attractor, if the sequence
(Bn)n∈N
satises
for every
n∈N
that
Bn∩K=∅
and if the measures given in item
(1)
and
(2)
are
self-similar, thanks to the estimates of the essential Hausdor content given by [9,
Theorem 2.6], Theorem 3.1 can be reformulated as follows:
•
In item
(1)
,
Hs
∞(e
Un∩lim supp→+∞Bp)
can be replaced by
Hs
∞(e
Un∩K)
or
Hµ,s
∞(e
Un)
and by
Hs
∞(e
Un)
when
µ
is the Lebesgue measure.
•
In item
(2),
one can replace
Hµ,s
∞(e
Un)
by
Hs
∞(e
Un∩K).
In particular the following corollary holds.

AN UPPER-BOUND FOR THE HAUSDORFF DIMENSION OF LIMSUP SETS 7
Corollary 3.3.
Let
S
be a self-similar IFS, let
µ
be a self-similar measure and
K
its attractor. Let
(Bn)n∈N
be a sequence of balls satisfying that
|Bn| → 0
and let
(Un)n∈N
be a sequence of open sets such that for every
n∈N, Un⊂Bn
. Assume
that
µ(lim supn→+∞Bn) = 1.
Dene
(e
Un)n∈N
as in
(8)
and consider
0≤s≤d.
(1) If for every
ε > 0
, for every
n∈N
large enough,
(11)
Hµ,s+ε
∞(e
Un)≤µ(3Bn)≤ Hµ,s−ε
∞(e
Un),
then
dimH(lim supn→+∞Un) = s.
(2) Assume that
(Bn)n∈N
is weakly redundant, that
(Bn)n∈N
is
µ
-a.c. and
that
(11)
holds with
Un
instead of
f
Un
and
µ(Bn)
instead of
µ(3Bn).
Then
dimHlim supn→+∞Un=s
.
In the case of the Lebesgue measure, Corollary 3.3 can simply be rephrased in
the following manner.
Corollary 3.4.
Let
(Bn)n∈N
a sequence of balls of
[0,1]d
satisfying
|Bn| → 0
and
(Un)n∈N
a sequence of open sets satisfying for every
n∈N, Un⊂Bn.
Assume that
Ld(lim supn→+∞Bn) = 1.
Dene
(e
Un)n∈N
as in
(8)
and consider
0≤s≤d.
(1)
If for every
ε > 0
, for every
n∈N
large enough,
(12)
Hs+ε
∞(e
Un)≤ Ld(Bn)≤ Hs−ε
∞(e
Un),
then
dimH(lim supn→+∞Un) = s.
(2)
If
(Bn)n∈N
is weakly redundant and if
(12)
holds with
Un
instead of
e
Un
.
Then
dimH(lim supn→+∞Un) = s.
As mentioned above, Corollaries 3.4 and 3.3 should play an important role in
situations where the balls
(Bn)
of comparable radii do overlap consequently. For
instance, given
S
a self-similar IFS and
K
its attractor, the shrinking targets asso-
ciated with
S
have been studied in [7] under the assumption
dimH(K) = dim(S),
where
dim(S)
is the similarity dimension of
S
. It is known that in many situations,
dimH(K) = min {d, dim(S)}
(see [16]), which raises the natural question of the
dimension of self-similar shrinking targets under the other natural assumption that
dim(S)> d
and
dimH(K) = min {d, dim(S)}=d.
In such situation, one expects
to have many overlaps. This will be the object of future investigations. Nonethe-
less, a toy example is provided in Section 5 to show that in many situations, one
should be able to estimate the contents involved in Corollaries 3.3 and 3.4.
In the next section, we apply Theorems 2.2 and 3.1 to study the sharpness of
the bounds provided by mass transference principles.
3.2.
Study of the optimality of mass transference principles for self-
similar measures.
As mentioned above, in this section, one studies the opti-
mality of mass transference principles from ball to ball and from ball to rectangles
in self-similar settings in an appropriate sense.Let us mention that, for the sake
of simplicity, the results of this section are stated for self-similar IFS but remain
valid when the underlying IFS is weakly conformal.
Corollary 3.5.
Let
µ∈ M(Rd)
be a self-similar measure of support
K
and
B=
(Bn)n∈N
be a sequence of balls centered in
K
satisfying
|Bn| → 0.

8 E. DAVIAUD
Assume that
B
is weakly redundant, that
µlim supn→+∞Bn= 1
and that
lim sup
n→+∞
log µ(Bn)
log(|Bn|)= dim(µ),
then for every
δ≥1
,
dimH(lim supn→+∞Bδ
n) = dim(µ)
δ.
In the case of mass transference from ball to rectangles, we obtain the following
corollary.
Corollary 3.6.
Let
µ
be a self-similar measure verifying that its support,
K
,
is the closure of its interior. Let
1≤τ1≤... ≤τd
,
τ= (τ1, ..., τd)
and let
(Bn:= B(xn, rn))n∈N
be a sequence of balls of
Rd
satisfying
rn→0.
Dene
Rn=˚
Rτ(xn, rn),
where
Rτ(xn, rn) = xn+Qd
i=1[−1
2rτi
n,1
2rτi
n].
Assume that
B
is weakly redundant, that
µlim supn→+∞Bn= 1
and that
lim sup
n→+∞
log µ(Bn)
log(|Bn|)= dim(µ),
then
dimH(lim supn→+∞Rn) = min1≤i≤dndim(µ)+P1≤j≤iτi−τj
τio.
By Theorem 2.2, given a self-similar measure satisfying the hypothesis of Corol-
laries 3.5 or 3.6, any sequence of balls
(Bn)n∈N
centered on
supp(µ)
satisfying
µlim supn→+∞1
2Bn= 1
admits a
µ
-a.c. sub-sequence of balls which veries the
hypotheses of the corollaries above so that the Hausdor dimension of the limsup
set associated with the corresponding
Un
's is provided by given corollaries. This
in particular proves that these bounds are sharp.
Remark 3.7.

Corollaries 3.5 and 3.6 are direct consequences of second
item of Remark 5.1 and Remark 5.3 in
[9]
, together with Corollary 3.3
(applied to
s=dim(µ)
δ
and
s= min1≤i≤dndim(µ)+P1≤j≤iτi−τj
τio
).

In the case of the Lebesgue measure, one always has
limn→+∞log µ(Bn)
log |Bn|=
dim(µ) = d.
As a consequence, when
Ld(lim supn→+∞Bn) = 1
,
(Un)
is a
sequence of balls or rectangles, the dimension of
lim supn→+∞Un
is given by
Corollaries 3.5 and 3.6 as soon as the sequence
(Bn)
is weakly redundant.
4.
Proof of Theorem 2.2
The concept of conditioned ubiquity was introduced by Barral and Seuret in
[2]. It consists in imposing that the balls of the sequence
(Bn)n∈N
verify some
specic properties with respect to a measure
µ
. As observed in many situations,
when a sequence of balls satises specic properties with respect to a measure
µ
, one can sometimes establish upper-bounds for
dimHlim supn→+∞Un.
Following
this observation, in the next section we prove Theorem 2.2, which establishes that,
under light assumptions on the sequence
(Bn)n∈N
, up to a
µ
-a.c. extraction, the
sequence
(Bn)n∈N
can always be assumed to satisfy convenient properties.
In this section, the balls
(Bn)n∈N
are supposed to be pairwise distinct
and such that
|Bn| →
n→+∞0
.

AN UPPER-BOUND FOR THE HAUSDORFF DIMENSION OF LIMSUP SETS 9
Theorem 2.2 is obtained by proving rst that it is always possible to extract
µ
-a.c weakly redundant sequences of balls. Then one proves in parallel that it is
also possible to extract
µ
-a.c sequences of balls which verify (6). The two next
sub-sections are dedicated to these results.
4.1.
Extraction of weakly redundant
µ
-a.c subsequences.
In the rst sub-
section, we establish the main extraction and the second sub-section, we give more
useful and insightful properties of weakly redundant sequences of balls.
4.1.1.
Proof of the main result.
The main result of this section is the following.
Proposition 4.1.
Let
µ∈ M(Rd)
and
(Bn)n∈N
be a
µ
-a.c sequence of balls. There
exists a subsequence
(Bψ(n))n∈N
of
(Bn)n∈N
which is weakly redundant and
µ
-a.c.
Proof.
Let us recall the following lemma.
Lemma 4.2
([8])
.
Let
µ∈ M(Rd)
and
B= (Bn:= B(xn, rn))n∈N
be a
µ
-a.c
sequence of balls of
Rd
with
limn→+∞rn= 0
.
Then for every open set
Ω
and every integer
g∈N
, there exists a subsequence
(B(Ω)
(n))⊂ {Bn}n≥g
such that:
(1)
∀n∈N
,
B(Ω)
(n)⊂Ω,
(2)
∀1≤n1=n2
,
B(Ω)
(n1)∩B(Ω)
(n2)=∅
,
(3)
µÄSn≥1B(Ω)
(n)ä=µ(Ω).
In addition, there exists an integer
NΩ
such that for the balls
(B(Ω)
(n))n=1,...,NΩ
, the
conditions (1) and (2) are realized, and (3) is replaced by
µÄSNΩ
n=1 B(Ω)
(n)ä≥3
4µ(Ω).
Let
gk∈N
be large enough so that
∀n≥gk
,
|Bn| ≤ 2−k.
By Lemma 4.2, applied
with the sequence
(Bn)n∈N
,
Ω = Rd
for any
k∈N
, there exists a sub-sequence
(B(n,k))
of
{Bn}N
n≥gk
satisfying
(1)
∀1≤n1=n2
,
B(n1,k)∩B(n2,k)=∅,
(2)
µSn∈NB(n,k)= 1.
Dene
Bψ= (Bψ(n))n∈N
as the sub-sequence of balls arising from
Sk∈NB(n,k)n∈N.
Since the following inclusion holds
(13)
\
k∈N[
n∈N
B(n,k)⊂lim sup
n→+∞
Bψ(n),
by item
(2)
one has
µ(lim sup
n→+∞
Bψ(n)) = 1.
Note that, for all
k∈N
, for all
B∈B(n,k)n∈N
,
|B| ≤ 2−k
. Following the
notation of Denition 2.4, for any
k∈N
,
Tk(Bψ)
can contain only balls of the
sequence of the
k
rst families
B(n,k)n∈N
, which are composed of pairwise disjoint
balls. This proves that
Tk(Bψ)
can be sorted in at most
k+ 1
families of pairwise
disjoint balls. In particular,
Bψ
is weakly redundant.
It remains to show that
(Bψ(n))n∈N
is
µ
-a.c.
Let
Ω
be an open set and
g∈N
. One will extract from
Bψ
a nite number of
balls satisfying the condition of Denition 2.3.

10 E. DAVIAUD
There exists
k0
so large that





µx:B(x, 2−k0+1)⊂Ω≥3µ(Ω)
4
for every
k≥k0,B(n,k)n∈N⊂Bψ(n)n≥g
µ(lim supn→+∞Bψ(n)∩Ω) ≥3µ(Ω)
4.
Setting
“
E=ßx∈lim sup
n→+∞
Bψ(n)∩Ω : B(x, 2−k0+1)⊂Ω™,
it holds that
µ(“
E)≥1
2µ(Ω).
Recalling (13), for every
x∈“
E
, consider
Bx
, the ball of
B(n,k0)n∈N
containing
x
. Note that, since for
B∈B(n,k0)n∈N
,
|B| ≤ 2−k0
, one has
Bx⊂B(x, 2−k0+1)⊂Ω.
Set
F1=¶Bx:x∈“
E©.
The set
F1
is composed of pairwise disjoint balls (by item
(1)
above) of
Bψ(n)n≥g
included in
Ω
and such that
(14)
µ[
L∈F1
L≥µ(“
E)≥1
2µ(Ω).
The
σ
-additivity of
µ
concludes the proof.
□
4.1.2.
More on weakly redundant sequences.
Recalling Denition 2.4, a sequence of
balls
(Bn)n∈N
is weakly redundant when at each scale
2−k
, the balls of the family
{Bn}n∈N
that have radii
≈2−k
can be sorted in a relatively small number of
families of pairwise disjoint balls. Its worth mentioning that if a sequence of balls
(Bn)n∈N
is weakly redundant, then for any
w > 1,
so is the sequence
(wBn)n∈N
(this fact is easily deduced from Lemma 5.5 below).
Remark 4.3.
Let us provide some examples of weakly redundant sequences of balls.

The sequence of rational balls
(B(p
q,1
q2))q∈N∗,0≤p≤q,p∧q=1
is weakly redundant
[3]
.

The sequence of closed dyadic cubes
(D)D∈Sn≥0Dn
is weakly redundant.

Let
(Xn)n∈N
be a sequence of i.i.d uniformly distributed random variables
on
[0,1]
. The sequence of balls
(B(Xn,1
n))n∈N
is weakly redundant
[3]
.

Let
m≥2
be an integer and
S={f1, ..., fm}
a self-similar IFS on
R
,
K
its attractor,
0< c1, ..., cm<1
the contraction ratios of
f1, ..., fm
,
Λ =
{1, ..., m}
and
Λ∗=Sn≥0Λn.
Given
n∈N
and
i= (i1, ..., in)∈ΛN
, we set
fi=fi1◦... ◦fin.
Let us recall that the similarity dimension of
S
,
dim(S),
is dened as the unique real number solution to the equation
m
X
i=1
cdim(S)
i= 1.

AN UPPER-BOUND FOR THE HAUSDORFF DIMENSION OF LIMSUP SETS 11
Assume that
S
satises the exponential separation condition, that is,
there exists
0< c < 1
such that for every
n∈N,
every
z∈K
, every
(i1, ..., in)= (j1, ..., jn)∈Λn
|fi1◦... ◦fin(z)−fj1◦... ◦fjn(z)| ≥ cn.
Then, due to a result of Barral and Feng
[1, Theorem 1.3]
, for every
z∈
K
the sequence
(B(fi(z),|fi(K)|))i∈Λ∗
is weakly redundant if and only if
dim(S)≤1.
The following proposition illustrate why the weak redundancy property relates
to overlaps between balls of the sequence of about the same radii.
Proposition 4.4.
Let
B= (Bn)n∈N
be a sequence of balls of
Rd.
Then the following
assertions are equivalent:
(1) The sequence
B
is weakly redundant.
(2) The sequence
B
satises
lim
n→+∞
log maxB∈Tn(B)#{B′∈ Tn(B) : B′∩B=∅}
n= 0.
Proof.
We show rst
(1) ⇒(2).
Fix
ε > 0.
Since
B
is assumed to be weakly redundant, there exists
nε∈N
large
enough so that for every
n≥nε,
log(Jn)
n≤ε.
Since for every
1≤j≤Jn
the family
Tn,j
is composed by pairwise disjoint balls,
by Lemma 5.4, there exists a constant
C > 0
depending on the dimension only
such that any ball
B∈ Tn(B)
intersects less than
C
balls of
Tn,j (B).
In particular,
(15)
log # {B′∈ Tn(B) : B′∩B=∅}
n≤log(CJn)
n≤ε+log C
n.
Since
C
does not depend on
B
, (15) holds for every
B
provided that
n
is large
enough, letting
ε→0,
one gets
lim
n→+∞
log maxB∈Tn(B)#{B′∈ Tn(B) : B′∩B=∅}
n= 0.
Let us now prove that
(2) ⇒(1)
.
Assume that
lim
n→+∞
log maxB∈Tn(B)#{B′∈ Tn(B) : B′∩B=∅}
n= 0.
Let
n∈N
and set
b
Jn= max
B∈Tn(B)#{B′∈ Tn(B) : B′∩B=∅} + 1.
Note that every
B∈ Tn(B)
intersects less than
b
Jn−1
balls of
Tn(B)
. Proceeding
as in the proof of [8, Lemma 3.2], one can sort the balls of
Tn(B)
in
b
Jn
families
Fn,1, ..., Fn, b
Jn
such that each family
Fi
is composed of pairwise disjoint balls. Since
lim
n→+∞
log b
Jn
n= 0,
the sequence
B
is weakly redundant.
□

12 E. DAVIAUD
The following lemma will be useful in the rest of the article when dealing with
weakly redundant sequences of balls.
Lemma 4.5.
Let
B= (Bn)n∈N
be a weakly redundant sequence of balls of
Rd
.
Then for every
µ∈ M(Rd)
and any
ε > 0,
one has
(16)
X
n∈N
|Bn|εµ(Bn)<+∞.
Proof.
Let
n∈N
and
Tn(B)
,
Jn
and
Tn,1(B), ..., Tn,Jn(B)
as in Denition 2.4. One
has
X
n∈N
|Bn|εµ(Bn) = X
n≥0X
1≤j≤JnX
B∈Tn,j (B)
|B|εµ(B)
≤X
n≥0X
1≤j≤Jn
2−nε X
B∈Tn,j (B)
µ(B).
Since for every
n∈N
and every
1≤j≤Jn,
the family
Tn,j (B)
is composed of
pairwise disjoint balls, one has
X
B∈Tn,j (B)
µ(B)≤1.
This, recalling that
log2Jn
n→0,
implies that
X
n∈N
|Bn|εµ(Bn)≤X
n≥0
2−nεJn<+∞.
□
4.2.
Extraction of sub-sequences of balls with conditioned measure.
Let
µ∈ M(Rd)
and
(Bn)n∈N
be a
µ
-a.c sequence of balls.
This part aims to understand what condition can be assumed about the measure
of the ball of the sequence
(Bn)n∈N
in general under the
µ
-a.c condition.
More precisely, item
(2)
of Theorem 2.2 is proved.
Proposition 4.6.
Let
µ∈ M(Rd)
. For any sequence of balls
(Bn)n∈N
satisfying
µ(lim supn→+∞vBn)=1
for some
0< v < 1
, there exists an
µ
-a.c sub-sequence
(Bϕ(n))n∈N
verifying
dimH(µ)≤lim inf
n→+∞
log µ(Bϕ(n))
log |Bϕ(n)|≤lim sup
n→+∞
log µ(Bϕ(n))
log |Bϕ(n)|≤dimP(µ).
Remark 4.7.
For the left part of
(4.6)
, the proof actually only uses the fact that
µ(lim supn→+∞Bn)=1.
Let us introduce some useful sets to prove Lemma 4.9 and Lemma 4.10, which
are key in order to prove (4.6).
Denition 4.1.
Let
0≤α≤γ
be real numbers,
µ∈ M(Rd)
, and
ε, ρ > 0
two
positive real numbers. Then dene
(17)
E[α,γ],ρ,ε
µ=¶x∈Rd: dim(µ, x)∈[α, γ]
and
∀r≤ρ, µ(B(x, r)) ≤rdim(µ,x)−ε©,
(18)
F[α,β],ρ,ε
µ=¶x∈Rd: dim(µ, x)∈[α, β]
and
∀r < ρ, µ(B(x, r)) ≥rdim(µ,x)+ε©.

AN UPPER-BOUND FOR THE HAUSDORFF DIMENSION OF LIMSUP SETS 13
and
E[α,γ],ε
µ=[
n≥1
E[α,γ],1
n,ε
µ
and
F[α,γ],ε
µ=[
n≥1
F[α,γ],1
n,ε
µ.
(19)
The following statements are easily deduced from Denition 2.2.
Proposition 4.8.
For every
µ∈ M(Rd)
,
ρ > 0
, every
0≤α≤γ
and
ε > 0
,
µ(E[α,γ],ε
µ) = µ({x: dim(µ, x)∈[α, γ]})
µ(F[α,γ],ε
µ) = µ(x: dim(µ, x)∈[α, γ])
and
E[α,γ],ρ,ε
µ⊂x∈Rd:∀r≤ρ, µ(B(x, r)) ≤rα−ε
(20)
F[α,γ],ρ,ε
µ⊂x∈Rd:∀r≤ρ, µ(B(x, r)) ≥rγ+ε.
Furthermore, for
α1= dimH(µ)
and
γ1=
supess
µ(dim(µ, x))
, one has
(21)
µ(E[α1,γ1],ε
µ)=1.
Similarly, for
α2=
infess
µ(dim(µ, x))
and
γ2= dimP(µ)
, one has
(22)
µÄF[α2,γ2],ε
µä= 1.
Proof.
For any
x∈Rd
, for any
ε > 0
, there exists
rx>0
such that,
∀r≤rx
,
rdim(µ,x)+ε≤µ(B(x, r)) ≤rdim(µ,x)−ε.
This implies
F[α,γ],ρ,ε
µ⊂x∈Rd:∀r≤ρ, µ(B(x, r)) ≥rγ+ε,
E[α,γ],ρ,ε
µ⊂x∈Rd:∀r≤ρ, µ(B(x, r)) ≤rα−ε,
and
E[α,γ],ε
µ={x: dim(µ, x)∈[α, γ]}
and
F[α,γ],ε
µ=x: dim(µ, x)∈[α, γ].
Since
µ([
infess
µ(dim(µ, x)),
supess
µ(dim(µ, x))]) = 1
and
µ([
infess
µ(dim(µ, x)),
supess
µ(dim(µ, x))]) = 1,
recalling Denition 2.2, it holds that, following the notation of Proposition 4.8,
µÄE[α1,γ1],ε
µä=µÄF[α2,γ2],ε
µä= 1.
□
Before showing Proposition 4.6, let us start by the two following Lemmas 4.9
and 4.10. The rst one will be used to prove the left part of the inequality (6)
while the second one will be useful to prove the right part.
Lemma 4.9.
Let
µ∈ M(Rd)
and
B= (Bn:= B(xn, rn))n∈N
be a
µ
-a.c sequence
of balls of
Rd
with
limn→+∞rn= 0
.
For any
ε > 0
, there exists a
µ
-a.c subsequence
(Bϕ(n))n∈N
of
B
such that for
every
n∈N
,
µ(Bϕ(n))≤(rϕ(n))dimH(µ)−ε.

14 E. DAVIAUD
Proof.
Set
α= dimH(µ)
and
γ=suppessµ(dim(µ, x)).
Let
Ω
be an open set and
ε > 0
. By (21),
µ(E[α,γ],ε
2
µ)=1
and
µ(Ω ∩E[α,γ],ε
2
µ) =
µ(Ω)
.
For every
x∈Ω∩E[α,γ],ε
2
µ
, there exists
rx>0
such that
B(x, rx)⊂Ω
and
x∈E[α,γ],rx,ε
2
µ.
Recall (19) and that the sets
E[α,γ],ρ,ε
µ
are non-increasing in
ρ
. In particular there
exists
ρΩ>0
such that the set
EΩ:= nx∈Ω∩E[α,γ],ρΩ,ε
2
µ:rx≤ρΩo
veries
(23)
µ(EΩ)≥3µ(Ω)
4.
Let
g∈N
. Applying Lemma 4.2 to
Ω
, the sequence
(Bn)
and the measure
m
,
there exists
NΩ
as well as
g≤n1≤... ≤nNΩ
verifying:
(1) for every
1≤i=j≤NΩ, Bni∩Bnj=∅
,
(2) for every
1≤i≤NΩ
,
2rni≤ρΩ
and
2α−ε
2≤r−ε
2
ni
,
(3)
µ(S1≤i≤NΩBni)≥µ(Ω)
2.
We may assume that
µ(Bni)>0
for every
i
, otherwise
Bni
does not play any role.
Item
(3)
together with (23) implies that
µ [
1≤i≤NΩ
Bni∩EΩ!≥µ(Ω)
4.
Furthermore, for every
1≤i≤NΩ
verifying
Bni∩EΩ=∅
, it holds that
0< µ(Bni)≤(rni)α−ε.
Indeed, let
x∈Bni∩EΩ
. By item (2),
Bni⊂B(x, 2rni)
,
and by (17) , item
(2)
, and (20), it holds that
µ(Bni)≤µ(B(x, 2rni)) ≤(2rni)α−ε
2≤(rni)α−ε.
Writing
B′={Bn:µ(Bn)≤rα−ε
n}
, the argument above shows that only balls
of
B′
have been used to cover
Ω
. This is satised for every open set
Ω
, so that
B′
is a sub-sequence of
B
satisfying the condition of Denition 2.3, which concludes
the proof of Lemma 4.9.
□
Lemma 4.10.
Let
µ∈ M(Rd)
,
v < 1
and
B= (Bn:= B(xn, rn))n∈N
a sequence
of balls of
Rd
verifying
µ(lim supn→+∞vBn) = 1
.
For all
ε > 0
, there exists a sub-sequence
(Bϕ(n))n∈N
of
B
as well as
0< v′<1
such that
µ(lim supn→+∞v′Bϕ(n)) = 1
and for all
n∈N
, one has
µ(Bϕ(n))≥
(rϕ(n))dimH(µ)+ε
.
Remark 4.11.
The sequence
(Bϕ(n))n∈N
found in Lemma 4.10 is in particular
µ
-a.c by Theorem 2.1.
Proof.
Let
α=
infess
µ(dim(µ, x))
and
γ= dimP(µ).
Let
ε > 0
and
v < v′<1
.
By (22) and Theorem 2.1,
µ(lim supn→+∞vBn∩F[α,γ],3ε
2
µ) = 1.
For all
x∈
lim supn→+∞vBn∩F[α,γ],3ε
2
µ
, there exists
rx>0
small enough so that
(24)
r
ε
2
x≤(v′−v)γ+3ε
2
and
∀0< r ≤rx, µ(B(x, r)) ≥rγ+3ε
2.
Since
x∈lim supn→+∞vBn
, for all
n∈N
, there exists
nx≥n
such that
x∈vBnx
and
(v′−v)rnx≤rx.
Note that
B(x, (v′−v)rnx)⊂v′Bnx.
This implies

AN UPPER-BOUND FOR THE HAUSDORFF DIMENSION OF LIMSUP SETS 15
the following inequalities:
µ(Bnx)≥µ(v′Bnx)≥µ(B(x, (v′−v)rnx)≥((v′−v)rnx)γ+3ε
2≥rγ+2ε
nx.
Set
Bγ,2ε={Bn:µ(Bn)≥rγ+2ε
n}
. One just showed that
lim sup
n→+∞
vBn∩F[α,γ],ε
2⊂lim sup
B∈Bγ,2ε
v′B.
This proves that
µ(lim supB∈Bγ,2εv′B) = 1
.
Since
ε > 0
was arbitrary, the results also holds with
ε
2
, which proves Lemma
4.10.
□
We are now ready to prove Proposition 4.6.
Proof.
Set
α= dimH(µ)
and
β= dimP(µ).
Let us x
(εn)n∈N∈(R∗+)N
verifying
limn→+∞εn= 0.
The strategy of the proof consists in constructing recursively coverings of the
cube
Rd
by using Lemma 4.9 and Lemma 4.10 and a diagonal argument (on the
choice of
ε
) at each step.
More precisely, at step 1, one will build a sequence of nite families of balls
(F1,i)i∈N
verifying:
(1) for all
i, j ≥1
,
∀L∈ F1,i
,
∀L′∈ F1,j
such that
L=L′,
one has
L∩L′=∅,
(2) for all
i≥1
,
F1,i
is a nite sub-family of
{Bn}n≥1,
(3) for all
i≥1
, for all
L∈ F1,i
,
|L|β+εi≤µ(L)≤ |L|α−εi,
(4) one has
(25)
µÑ[
i∈N[
L∈F1,i
Lé= 1.
At step 2, a family of balls
(F2,i)i∈N
will be constructed such that items
1,2,3
and
4
holds with
ε=εi+1.
Write
F2=Si≥1F2,i
.
The other steps are achieved following the same scheme.
The construction is detailed below:
Step 1:
Let
Ω1,1=Rd.
Sub-step 1.1:
By Lemma 4.9 and Lemma 4.10 applied to
ε=ε1
, there exists a
µ
-a.c sub-
sequence
(Bψ1,1(n))n∈N
, satisfying, for every
n∈N
,
|Bψ1,1(n)|β+ε1≤µ(Bψ1,1(n))≤ |Bψ1,1(n)|α−ε1.
By Lemma 4.2 applied to
Ω1,1
, the sequence
(Bψ1,1(n))n∈N
and
g= 1
, there exists
an integer
N1,1
as well as some balls
L1,1,1, ..., L1,1,N1,1∈ {Bn}n≥1
verifying:

for all
1≤i<j≤N1,1
,
L1,1,i ∩L1,1,j =∅,

for all
1≤i≤N1,1
,
|L1,1,i|β+ε1≤µ(L1,1,i )≤ |L1,1,i|α−ε1,

µ(S1≤i≤N1,1L1,1,i)≥1
2.

16 E. DAVIAUD
Set
F1,1={L1,1,i}1≤i≤N1,1.
Sub-step 1.2:
Let
Ω1,2= Ω1,1\SL∈F1,1L.
By Lemma 4.9 and Lemma 4.10 with
ε=ε2
, there exists a
µ
-a.c sub-sequence
(Bψ1,2(n))n∈N
satisfying
|Bψ1,2(n)|β+ε2≤µ(Bψ1,2(n))≤ |Bψ1,2(n)|α−ε2.
One applies Lemma 4.2 to the open set
Ω1,2
, the sub-sequence
(Bψ1,2(n))n∈N
and
g= 1
. There exists
N1,2∈N
such that
L1,2,1, ..., L1,2,N1,2
veries:

for all
1≤i<j≤N1,2
,
L1,2,i ∩L1,2,j =∅,

for all
1≤i≤N1,2
,
|L1,2,i|β+ε2≤µ(L1,2,i )≤ |L1,2,i|α−ε2,

µ(S1≤i≤N1,2L1,2,i)≥1
2µ(Ω1,2).
The family
F1,2
is dened as
F1,2={L1,2,i}1≤i≤N1,2.
Proceeding iteratively as in Sub-steps
1.1
and
1.2
, for any
i∈N
, at Sub-step
1.i
a nite family of balls
F1,i
is constructed so that the items
1,2,3
and
4
holds
with
εi
(instead of
ε1
).
Recall that, to justify the last item, this recursive scheme allows to cover
Rd
, up
to a set of
µ
-measure 0 (the argument is similar to the one developed at the end
of the proof of Lemma 4.2 in [8]).
Set
F1=Si≥1F1,i.
Let us notice that the construction of the family
F2
does not rely on the existence
of the family
F1
, so that the families
Fk
can actually be built independently,
following the same scheme, as described below.
Step
k
:
As in step 1, one constructs a family of balls
(Fk,i)i≥1
verifying items
1,2,3
and
4
with
ε=εk+i−1.
Set
F=[
k≥1
Fk
with
Fk=[
i≥1
Fk,i.
Denote by
(Bϕ(n))n∈N
the sub-sequence of balls that constitutes the family
F.
By construction, for all
i∈N
, denoting
Nk,i = #Fk,i,
for every
n∈N
there are at most
N≥P1≤i,k≤nNk,i
balls of
Bϕ(k)k∈N
belonging
to
S1≤i,k≤nFi,k.
As a consequence, for
‹
N
large enough and every
n′≥‹
N
, one has
|Bϕ(n′)|β+εn≤µ(Bϕ(n′))≤ |Bϕ(n′)|α−εn.
It follows that
α−εn≤lim inf
n′→+∞
log µ(Bϕ(n′))
log |Bϕ(n′)|≤lim sup
n′→+∞
log µ(Bϕ(n′))
log |Bϕ(n′)|≤β+εn.
Letting
n→+∞
shows that
dimH(µ)≤lim inf
n′→+∞
log µ(Bϕ(n′))
log |Bϕ(n′)|≤lim sup
n′→+∞
log µ(Bϕ(n′))
log |Bϕ(n′)|≤dimP(µ).
It only remains to prove that
(Bϕ(n))n∈N
is
µ
-a.c.

AN UPPER-BOUND FOR THE HAUSDORFF DIMENSION OF LIMSUP SETS 17
Let
Ω
be an open set and
g∈N
. We nd a nite family of balls
{L}i∈I ⊂
Bϕ(n)n≥g
satisfying the conditions of Denition 2.3.
Note that, by (25), setting
E=Tk≥1SL∈FkL,
one has
µ(E) = 1.
Let
x∈Ω∩
E
and
rx>0
small enough so that
B(x, rx)⊂Ω
and consider
kx≥ϕ(g)≥g
large
enough so that, for all
n≥kx
,
|Bn| ≤ 2rx
. Recall that
Fkx⊂ {Bn}n≥kx.
Finally,
let us x
k
large enough so that
µ(“
E)≥µ(Ω)
2,
where
“
E={x∈E:kx≤k}
. For
x∈“
E
, let
Lx∈ Fk
be the ball that contains
x
(the balls of
Fk
being pairwise
disjoint,
Lx
is well dened) and
{Li}i≥1=¶Lx:x∈“
E©
. One has

for all
1≤i<j
,
Li∩Lj=∅,

for all
i∈N
,
Li∈Bϕ(n)n≥g
and
Li⊂Ω,

µ(Si≥1Li)≥µ(“
E)≥µ(Ω)
2.
By
σ
-additivity, there exists
N∈N
such that
µ(S1≤i≤NLi)≥µ(Ω)
4
, which proves
that
(Bϕ(n))n∈N
satises Denition 2.3 with
C=1
4
and is indeed
µ
-a.c.
□
Proposition 4.1 and Proposition 4.6 together prove Theorem 2.2.
5.
Proof of Theorem 3.1
5.1.
Proof of item
(1)
of Theorem 3.1.
The proof of Theorem 3.1 will be
achieved by proving that the result stands for weakly redundant sequences of balls
and that the general case can indeed be deduced from this particular case (see
Lemma 5.7 below).
5.1.1.
Proof in the weakly redundant case.
Let
(Bn)n∈N
be a sequence of balls of
Rd
satisfying
|Bn| → 0
and
(Un)n∈N
a sequence of open sets such that for every
n∈N, Un⊂Bn.
Let us start by the following proposition.
Proposition 5.1.
Let
0≤s≤d.
Assume that
(Bn)n∈N
is weakly redundant and that there exists
µ∈ M(Rd)
as
well as a Borel set
A⊂Rd
such that
lim sup
n→+∞
Un⊂A
and
∀n∈N,Hs
∞(Un∩A)≤µ(Bn).
Then
dimH(lim sup
n→+∞
Un)≤s.
Proof.
For any
n∈N,
let
(Ak,n)k∈N
be a sequence of open balls such that,
|Ak,n| ≤
|Un|, Un∩A⊂Sk≥0Ak,n
and
(26)
X
k≥0
|Ak,n|s≤2Hs
∞(Un∩A)≤2µ(Bn).
Note that, since
Un∩A⊂Sk≥0Ak,n
, one has
lim sup
n→+∞
Un⊂lim sup
k,n→+∞
Ak,n.
Moreover, since
(Bn)n∈N
is weakly redundant (see Lemma 4.5),
X
k,n≥0
|Ak,n|s+ε≤X
n≥0
|Bn|εX
k≥0
|Ak,n|s≤2X
n≥0
|Bn|εµ(Bn)<+∞.

18 E. DAVIAUD
One concludes that
dimH(lim sup
n→+∞
Un)≤dimH(lim sup
n→+∞
Ak,n)≤s+ε.
Letting
ε
tend to
0
yields the desired conclusion.
□
Remark 5.2.
By taking
A= lim supp→+∞Bp
, one recovers Theorem 3.1 where
f
Un
is replaced by
Un
and
3Bn
is replaced by
Bn
.
5.1.2.
Proof in the general case.
Let us rst state a modied version of the cele-
brated
5r
-lemma which allows to drop the constant
5
to
(1 + ε).
Lemma 5.3
(
(1 + ε)r
-lemma)
.
For every
ε > 0
, there exists a constant depending
on
d
and
ε
,
Cd,ε
, such that for every family of balls
B
such that
supB∈B |B|<+∞,
there exists some families of balls
F1, ..., FCd,ε ⊂ B
satisfying:

for every
1≤i≤Cd,ε
, for every
B=B′∈ Fi
B∩B′=∅,

for every
B∈ B
there exists
L∈S1≤i≤Cd,ε Fi
such that
B⊂(1 + ε)L.
In particular, one has
[
B∈B
B⊂[
1≤i≤Cd,ε [
L∈Fi
(1 + ε)L.
Proof.
Let us recall rst the following lemma.
Lemma 5.4
([8])
.
For any
0< v ≤1
there exists a constant
γv,d >0
depending
only on
v
and the dimension
d
only, satisfying the following: if a family of balls
B= (Bn)n∈N
and a ball
B
are such that
• ∀ n≥1
,
|Bn| ≥ 1
2|B|,
• ∀ n1=n2≥1
,
vBn1∩vBn2=∅,
then
B
intersects at most
γv,d
balls of
B
.
Let
ε0>0
be small enough so that
(1 + ε0)2≤1 + ε.
We set
Cd,ε = (γd, ε0
2+ 1) ×(⌊−log ε0
log(1+ε0)⌋+ 2).
The families
F1, ..., FC(d,ε)
will be constructed recursively by adding balls at each
step to one of these families. For now we set for every
1≤i≤Cd,ε
,
Fi=∅.
For every
k≥0,
dene
Bk=ßB:supL∈B |L|
(1 + ε0)k+1 ≤ |B|<supL∈B |L|
(1 + ε0)k™.
Let
Gk
a maximal family of balls of
Bk
satisfying that, for every
L=L′∈ G0,
ε0
2L∩ε0
2L′=∅.
By maximality, for every
L′′ ∈ Bk
there exists
L∈ Gk
which satises satises
ε0
2L′′ ∩ε0
2L=∅.

AN UPPER-BOUND FOR THE HAUSDORFF DIMENSION OF LIMSUP SETS 19
This implies that
L′′ ⊂(1 + ε)L.
We describe now how we sort the balls in the dierent families.
Note that if a ball
L∈ Gk
intersects a ball
L′∈ Gk′
with
k′≤k− ⌊ −log ε0
log(1+ε0)⌋ − 1
,
then the radius of
L′
is so much larger than the radius of
L
that
L⊂(1 + ε)L′.
In the cases where balls of
Gk
intersects balls of
Gk′
for
k− ⌊ −log ε0
log(1+ε0)⌋ ≤ k′≤k,
let us explain how to proceed for the rst steps. We recall the following lemma
established in [8].
Lemma 5.5.
Let
0< v < 1
and
B= (Bn)n∈N
be a countable family of balls such
that
limn→+∞|Bn|= 0
, and for every
n=n′∈N
,
vBn∩vB′
n=∅
.
There exists
γd,v + 1
(
γd,v
being the constant appearing in Lemma 5.4 below)
sub-families of
B
,
(Fi)1≤i≤γd,v+1
, such that:
• B =S1≤i≤γd,v+1 Fi
,
• ∀ 1≤i≤γd,v + 1
,
∀L∩L′∈ Fi
, one has
L∩L′=∅.
By Lemma 5.5, for each
k≥0,
it is possible to sort the balls of
Gk
in at most
γd, ε0
2+ 1
families of pairwise disjoint balls. In particular we can sort the balls of
[
0≤k≤⌊ −log ε0
log(1+ε0)⌋+1
Gk
in
F1, ..., FCd,ε
.
Consider
L∈ G⌊−log ε0
log(1+ε0)⌋+2.
If there exists
L′∈ G0
such that
L′∩L=∅,
since
|L| ≤ ε(1 + ε) supL∈B |L|,
one has
L⊂(1 + ε)L′.
Otherwise,
L
intersects only balls of
S1≤k≤⌊ −log ε0
log(1+ε0)⌋+1 Gk
. For each
1≤k≤
⌊−log ε
log(1+ε0)⌋+1,
the set of balls of
Gk
which intersect
L
satises the hypotheses Lemma
5.4 with
B=L
so that
L
can not intersect more than
γd, ε0
2
such balls. In particular,
L
does not intersect more than
γd, ε0
2×(⌊−log ε0
log(1+ε0)⌋+ 1)
balls of
S1≤k≤⌊ −log ε0
log(1+ε0)⌋+1 Gk
.
Since
Cd,ε > γd, ε0
2×(⌊−log ε0
log(1+ε0)⌋+ 1)
, there must exist
1≤i≤Cd,ε
such that
L
does not intersect any ball of
Fi.
We add
L
to the family
Fi.
The rest of the proof readily follows from using this argument recursively on the
balls of
Gk
and on
k
.
□
Remark 5.6.
It is worth mentioning that the
5r
-lemma holds in any metric space
while Lemma 5.3 uses the fact that
Rd
is direction-limited
[13]
(and in particular,
this lemma does not hold in any metric space).
We now establish the general case of item
(1)
of Theorem 3.1.
Applying Lemma 5.3 to each family
Tk(B)
and
ε=1
4
, for
k∈N
, one gets the
following result.
Lemma 5.7.
There exists
C > 0
and a sub-sequence
Bϕ= (Bϕ(n))n∈N
satisfying
the following property:

20 E. DAVIAUD
(1)
for every
k∈N
and every
n∈N
such that
Bn∈ Tk(B),
there exists
n′∈N
such that
Bϕ(n′)∈ Tk(B)
and
Bn⊂3
2Bϕ(n′).
(2)
for every
n∈N
and
k∈N
such that
Bϕ(n)∈ Tk(B),
one has
#Bϕ(n′)∈ Tk(B) : Bϕ(n′)∩Bϕ(n)≤C.
Remark 5.8.
(1)
By item
(1)
of the proposition above, if there exists a mea-
sure
µ∈ M(Rd)
such that
µ(lim supn→+∞Bn)=1,
then one also has that
µ(lim sup
n→+∞
3
2Bϕ(n))=1
and, by Theorem 2.1,
(3Bϕ(n))n∈N
is
µ
-a.c.
(2)
By item
(2)
and Lemma 5.4, the sequence
(3Bϕ(n))n∈N
is weakly redundant.
(3)
By item
(1)
, for every
p∈N
, there exists
Bϕ(n)∈ Bϕ
such that
Bp⊂3Bϕ(n).
Recalling
(8)
, this implies that
Up⊂e
Uϕ(n).
In particular,
lim sup
n→+∞
Un= lim sup
n→+∞e
Uϕ(n)= lim sup
n→+∞e
Un.
We are now ready to nish the proof of Theorem 3.1.
Let
(Bϕ(n))
a sequence given by Lemma 5.7 and
0≤s≤d.
Assume that there
exists
µ∈ M(Rd)
such that, for any
n∈N,
Hs
∞(e
Un∩lim sup
p→+∞
Bp)≤µ(3Bn).
Applying Lemma 5.1 with
A= lim supp→+∞Bp
,
(e
Uϕ(n))n∈N
and
(3Bϕ(n))n∈N
, one
gets
dimH(lim sup
n→+∞e
Uϕ(n))≤s.
Item
(3)
of Remark 5.8 concludes the proof of Theorem 3.1.
5.2.
Proof of item
(2)
of Theorem 3.1.
Item
(2)
is in fact a direct application
of Theorem 5.9 below, to
(B′
n)n∈N= (3Bϕ(n))n∈N
and the sequence of open sets
(Vn)n∈N= ( e
Un)n∈N
together with item
(1)
of Remark 5.8.
Theorem 5.9
( [9])
.
Let
µ∈ M(Rd)
and
B= (B′
n)n∈N
be a
µ
-a.c. sequence of
closed balls of
Rd
. Let
(Vn)n∈N
be a sequence of open sets such that
Vn⊂B′
n
for
all
n∈N
, and
0≤s≤dimH(µ)
. If
lim sup
n→+∞
log Hµ,s
∞(Vn)
log µ(B′
n)≤1,
then
dimH(lim sup
n→+∞
Vn)≥s.

AN UPPER-BOUND FOR THE HAUSDORFF DIMENSION OF LIMSUP SETS 21
5.3.
A toy example: detecting sequences of too large balls.
Let
(Bn)n∈N
be a sequence of balls satisfying
|Bn| → 0
and
δ≥1.
As mentioned earlier in the
paper in the situation where
µ(lim supn→+∞Bn) = 1
for
µ∈ M(Rd)
a measure
carrying enough self-similarity (say the Lebesgue measure on
[0,1]d
for instance),
the usual bound for
dimH(lim supn→+∞Bδ
n)
might not be relevant when the balls
(Bn)n∈N
overlaps too much to begin with.
In this section, we show on a toy example how Theorem 3.1 allows to obtain
better bounds in such cases.
For
k∈N
, let us denote by
Dk
the set of dyadic cubes of
[0,1]d
of generation
k
and by
Sk
the set of dyadic numbers of generation
k
. We x
0< η < 1
and we
consider the sequence
(Bk,n)n∈N,0≤k≤2n−1
dened as
(27)
Bn,k =B(k
2n,1
2ηn ).
Let us x
δ≥1
and set
Un,k =Bδ
n,k.
It easily proved that
dimHlim sup
n∈N,0≤k≤2n−1
Un,k = min ßd, d
ηδ ™
but this is done by splitting between the cases
1≤δ≤1
η
and
δ≥1
η.
We prove
here that one recovers the right bound by applying only Corollary 3.4.
Set
(28)
e
Un,k =[
m,p∈N:Bm,p⊂3Bn,k
and
1
2≤|Bn,k|
|Bm,p|≤2
Um,p.
Proposition 5.10.
There exists
C > 0
such that for every
n∈N
and for every
0≤k≤2n−1
,
(29)
C−1Ld(Bn,k)≤ Hmin{d
ηδ ,d}
∞(e
Un,k)≤CLd(Bn,k )
Before proving this proposition, let us show that Proposition 5.10 together with
Corollary 3.4 allow to compute
dimHlim supn∈N,0≤k≤2−n−1Un,k = min ¶d, d
ηδ ©
.
Remark rst that
Ld(lim supn∈N,0≤k≤2n−1Bn,k) = 1.
Moreover by Proposition
5.10, for
s= min ¶d, d
ηδ ©
, for every
ε > 0
and
n∈N
large enough,
Hs+ε
∞(e
Un,k)≤ Ld(Bn,k )≤ Hs−ε
∞(e
Un,k).
By application of Corollary 3.4 with
(Bn) = (Bn,k)
and
(Un) = (Un,k =Bδ
n,k)
, one
gets
dimHlim sup
n∈N,0≤k≤2−n−1
Bδ
n,k = min ßd, d
ηδ ™.
We now establish Proposition 5.10.
Proof.
Note rst that if
0≤δ < 1
η,min ¶d
ηδ , d©=d
and
Bn,k ⊂e
Un,k ⊂3Bn,k
so
that
C−1Ld(Bn,k)≤ Hd
∞(Bn,k)≤ Hd
∞(e
Un,k)≤ Hd
∞(3Bn,k)≤CLd(Bn,k ).
Assume now that
δ≥1
η
and let us rst establish the upper-bound.

22 E. DAVIAUD
We x
0≤s≤d
ηδ
and we consider the two coverings of
e
Un,k
C1={3Bn,k}
and
C2=[
m,p∈N:Bm,p⊂3Bn,k
and
1
2≤|Bn,k|
|Bm,p|≤2
{Um,p}.
A counting argument shows that there exists
κ > 0
such that for
i= 1,2,
(30)
Hs
∞(e
Un,k)≤min
i∈{1,2}X
A∈Ci
|A|s≤κ·2−kη max{s,d(1−1
η)+sδ}.
Note that
s≤d(1 −1
η) + sδ ⇔s≥d
1
η−1
δ−1
and
d(1 −1
η) + sδ ≤d⇔s≤d
ηδ .
Since
δ≥1
η,
one has
d
ηδ ≥d
1
η−1
δ−1,
so that
max ßs, d(1 −1
η) + sδ™=d(1 −1
η) + sδ.
By taking
s=d
ηδ
, one gets
Hs
∞(e
Un,k)≤κ2−kηd ≤CLd(Bn,k ).
We now show that the left-hand side of (29) holds.
Set
F(Bn,k) = ßm, p ∈N:Bm,p ⊂3Bn,k
and
1
2≤|Bn,k|
|Bm,p|≤2™
and consider the measure
µ∈ M(Rd)
dened by
(31)
µ(·) = Pm,p∈N:Bm,p ⊂3Bn,k
and
1
2≤|Bn,k|
|Bm,p|≤2
Ld(Um,p∩·)
Ld(Um,p)
#F(Bn,k).
Lemma 5.11.
There exists a constant
C0>0
such that, for any ball
A
of
[0,1]d
and any
0≤s≤d
, one has
(32)
µ(A)≤C0
|A|s
2−kη(d(1−1
η)+sδ).
Proof.
Note that there exists a universal constant
C > 0
such that
C−1≤#F(Bn,k)
|Bn,k|d(1−1
η)≤C.
Let
A
be a ball with
|A| ≤ |Bn,k|.
Fix
t≥1
such that
|A|=|Bn,k|t.

if
1≤t≤1
η
:
there exists a universal constant
C1
such that
A
intersects
less than
C1×|A|d
|Bn,k|d
η
balls of
F(Bn,k).
This gives
µ(A)≤CC1×|A|d
|Bn,k|d
η
×|Bn,k|d
η
|Bn,k|d=C C1Å|A|
|Bn,k|ãd−s
|Bn,k|−s× |A|s
(33)
≤CC1|Bn,k|−s× |A|s.

AN UPPER-BOUND FOR THE HAUSDORFF DIMENSION OF LIMSUP SETS 23

if
1
η≤t≤δ
:
the ball
A
intersects at most
C2
balls of
F(Bn,k)
(where
C2
is a constant that depends on
η
and
d
). In particular,
µ(A)≤C2C|Bn,k|d
η
|Bn,k|d=C2C|Bn,k |d
η
|Bn,k|d|A|s|A|s.
Since
|A|=|Bn,k|t≥ |Bn,k |δ,
one has
1
|A|s≤1
|Bn,k|sδ .
This gives
(34)
µ(A)≤CC2|Bn,k|d(1
η−1)−sδ|A|s.

If
t>δ:
the ball
A
intersects at most one of the ball of
F(Bn,k)
, so that
µ(A)≤C|Bn,k|d
η
|Bn,k|d×|A|d
|Bn,k|d
δ
=CÅ|A|
|Bn,k|δãd−s
× |Bn,k|d(1
η−1)−sδ × |A|s
(35)
≤C|Bn,k|d(1
η−1)−sδ × |A|s.
(36)
By (33), (34) and (35), there exists a universal constant
C0>0
such that, for any
ball
A
with
|A|≤|Bn,k|,
µ(A)≤C02kη(d(1−1
η)+sδ)|A|s,
which was the statement.
□
Recall that
µ(e
Un,k) = 1.
Recalling 5.11, this implies that
Hs
∞(e
Un,k)≥1
C02kη(d(1−1
η)+sδ)=2−kη(d(1−1
η)+sδ)
C0
.
Taking
s=d
ηδ ,
one gets
Hs
∞(e
Un,k)≥2−kdη
C0
≥CLd(Bn,k),
which concludes the proof of Proposition 5.10.
□
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