A COMBINATORIAL PROOF OF MARSTRAND'S THEOREM FOR ...

A COMBINATORIAL PROOF OF MARSTRAND’S THEOREM
FOR PRODUCTS OF REGULAR CANTOR SETS
YURI LIMA AND CARLOS GUSTAVO MOREIRA
Abstract. In a paper from 1954 Marstrand proved that if K ⊂ R 2 has Hausdorff
dimension greater than 1, then its one-dimensional projection has positive
Lebesgue measure for almost-all directions. In this article, we give a combinatorial
proof of this theorem when K is the product of regular Cantor sets of
class C 1+α , α > 0, for which the sum of their Hausdorff dimension is greater
than 1.
1. Introduction
If U is a subset of R n , the diameter of U is |U| = sup{|x−y|;x,y ∈ U} and, if
U is a family of subsets of R n , the diameter of U is defined as
‖U‖ = sup |U|.
U∈U
Given d > 0, the Hausdorff d-measure of a set K ⊆ R n is
)
∑
m d (K) = lim |U| d
ε→0
(
inf
U covers K
‖U‖ d 0 . We define the Hausdorff
dimension of K as HD(K) = d 0 . Also, for each θ ∈ R, let v θ = (cosθ,sinθ), L θ
the line in R 2 through the origin containing v θ and proj θ : R 2 → L θ the orthogonal
projection. From now on, we’ll restrict θ to the interval [−π/2,π/2], because L θ =
L θ+π .
In 1954, J. M. Marstrand [4] proved the following result on the fractal dimension
of plane sets.
Theorem. If K ⊆ R 2 is a Borel set such that HD(K) > 1, then m(proj θ (K)) > 0
for m-almost every θ ∈ R.
The proof is based on a qualitative characterization of the “bad” angles θ for
which the result is not true. Specifically, Marstrand exhibits a Borel measurable
function f(x,θ), (x,θ) ∈ R 2 ×[−π/2,π/2], such that f(x,θ) = +∞ for m d -almost
every x ∈ K, for every “bad” angle. In particular,
∫
f(x,θ)dm d (x) = +∞. (1.1)
K
U∈U
On the other hand, using a version of Fubini’s Theorem, he proves that
∫ π/2 ∫
dθ f(x,θ)dm d (x) = 0
−π/2
K
Key words and phrases. Cantor sets, Hausdorff dimension, Marstrand theorem.
1
.

2 YURI LIMA AND CARLOS GUSTAVO MOREIRA
which, in view of (1.1), implies that
m({θ ∈ [−π/2,π/2]; m(proj θ (K)) = 0}) = 0.
These results are based on the analysis of rectangular densities of points.
Many generalizationsand simpler proofs haveappeared since. One of them came
in 1968 by R. Kaufman who gave a very short proof of Marstrand’s theorem using
methods of potential theory. See [2] for his original proof and [5], [9] for further
discussion.
In this article, we prove a particular case of Marstrand’s Theorem.
Theorem 1.1. If K 1 ,K 2 are regular Cantor sets of class C 1+α , α > 0, such that
d = HD(K 1 ) + HD(K 2 ) > 1, then m(proj θ (K 1 ×K 2 )) > 0 for m-almost every
θ ∈ R.
The argument also works to show that the push-forward measure of the restriction
of m d to K 1 ×K 2 , defined as µ θ = (proj θ ) ∗ (m d | K1×K 2
), is absolutely continuous
with respect to m, for m-almost every θ ∈ R. Denoting its Radon-Nykodim
derivative by χ θ = dµ θ /dm, we also prove the following result.
Theorem 1.2. χ θ is an L 2 function for m-almost every θ ∈ R.
Remark1.3. Theorem1.2,asinthiswork,followsfrommostproofsofMarstrand’s
theorem and, in particular, is not new as well.
Our proof makes a study on the fibers proj θ −1 (v)∩(K 1 ×K 2 ), (θ,v) ∈ R×L θ ,
and relies on two facts:
(I) A regular Cantor set of Hausdorff dimension d is regular in the sense that the
m d -measure of small portions of it has the same exponential behavior.
(II)Thisenablesustoconcludethat, exceptforasmallsetofanglesθ ∈ R, thefibers
proj θ −1 (v) ∩(K 1 ×K 2 ) are not concentrated in a thin region. As a consequence,
K 1 ×K 2 projects into a set of positive Lebesgue measure.
The idea of (II) is based on the work [6] of the second author. He proves that,
if K 1 and K 2 are regular Cantor sets of class C 1+α , α > 0, and at least one
of them is non-essentially affine (a technical condition), then the arithmetic sum
K 1 +K 2 = {x 1 +x 2 ;x 1 ∈ K 1 ,x 2 ∈ K 2 } has the expected Hausdorff dimension:
HD(K 1 +K 2 ) = min{1,HD(K 1 )+HD(K 2 )}.
Marstrand’s Theorem for products of Cantor sets has many useful applications
in dynamical systems. It is fundamental in certain results of dynamical bifurcations,
namely homoclinic bifurcations in surfaces. For instance, in [10] it is used
to show that hyperbolicity is not prevalent in homoclinic bifurcations associated to
horseshoes with Hausdorff dimension larger than one; in [7] it is used to prove that
stable intersections of regular Cantor sets are dense in the region where the sum of
their Hausdorff dimensions is larger than one; in [8] to show that, for homoclinic
bifurcations associated to horseshoes with Hausdorff dimension larger than one,
typically there are open sets of parameters with positive Lebesgue density at the
initial bifurcation parameter corresponding to persistent homoclinic tangencies.
2. Regular Cantor sets of class C 1+α
We say that K ⊂ R is a regular Cantor set of class C 1+α , α > 0, if:
(i) there are disjoint compact intervals I 1 ,I 2 ,...,I r ⊆ [0,1] such that K ⊂
I 1 ∪···∪I r and the boundary of each I i is contained in K;

MARSTRAND’S THEOREM FOR PRODUCTS OF CANTOR SETS 3
(ii) there is a C 1+α expanding map ψ defined in a neighbourhood of I 1 ∪I 2 ∪
···∪I r such that ψ(I i ) is the convex hull of a finite union of some intervals
I j , satisfying:
(ii.1) for each i ∈ {1,2,...,r} and n sufficiently big, ψ n (K ∩I i ) = K;
(ii.2) K = ⋂ ψ −n (I 1 ∪I 2 ∪···∪I r ).
n∈N
The set {I 1 ,...,I r } is called a Markov partition of K. It defines an r×r matrix
B = (b ij ) by
b ij
= 1, if ψ(I i ) ⊇ I j
= 0, if ψ(I i )∩I j = ∅,
which encodes the combinatorial properties of K. Given such matrix, consider the
set Σ B = { θ = (θ 1 ,θ 2 ,...) ∈ {1,...,r} N ; b θiθ i+1
= 1,∀i ≥ 1 } and the shift transformation
σ : Σ B → Σ B given by σ(θ 1 ,θ 2 ,...) = (θ 2 ,θ 3 ,...).
There is a natural homeomorphism between the pairs (K,ψ) and (Σ B ,σ). For
each finite word a = (a 1 ,...,a n ) such that b aia i+1
= 1, i = 1,...,n − 1, the
intersection
I a = I a1 ∩ψ −1 (I a2 )∩···∩ψ −(n−1) (I an )
is a non-empty interval with diameter |I a | = |I an |/|(ψ n−1 ) ′ (x)| for some x ∈ I a ,
which is exponentially small if n is large. Then, {h(θ)} = ⋂ n≥1 I (θ 1,...,θ n) defines a
homeomorphism h : Σ B → K that commutes the diagram
Σ B
σ
Σ B
h
h
K ψ
K
Ifλ = sup{|ψ ′ (x)|;x ∈ I 1 ∪···∪I r } ∈ (1,+∞), then ∣ I(θ1,...,θ ∣
n+1) ≥ λ −1·∣∣ I(θ1,...,θ ∣ n)
and so, for ρ > 0 small and θ ∈ Σ B , there is a positive integer n = n(ρ,θ) such that
ρ ≤ ∣ I(θ1,...,θ ∣ n) ≤ λρ.
Definition 2.1. A ρ-decomposition of K is any finite set (K) ρ = {I 1 ,I 2 ,...,I r } of
disjoint closed intervals of R, each one of them intersecting K, whose union covers
K and such that
ρ ≤ |I i | ≤ λρ, i = 1,2,...,r.
Remark 2.2. Although ρ-decompositions are not unique, we use, for simplicity,
the notation (K) ρ to denote any of them. We also use the same notation (K) ρ to
denote the set ∪ I∈(K)ρ I ⊂ R and the distinction between these two situations will
be clear throughout the text.
EveryregularCantorset ofclass { C 1+α has a ρ-decomposition } for ρ > 0 small: by
the compactness of K, the family I (θ1,...,θ n(ρ,θ))
has a finite cover (in fact, it
θ∈Σ B
is only necessary for ψ to be of class C 1 ). Also, one can define ρ-decomposition for
the product of two Cantor sets K 1 and K 2 , denoted by (K 1 ×K 2 ) ρ . Given ρ ≠ ρ ′
and two decompositions (K 1 ×K 2 ) ρ ′ and (K 1 ×K 2 ) ρ , consider the partial order
⋃ ⋃
(K 1 ×K 2 ) ρ ′ ≺ (K 1 ×K 2 ) ρ ⇐⇒ ρ ′ < ρ and Q ′ ⊆
Q ′ ∈(K 1×K 2) ρ ′
Q∈(K 1×K 2) ρ
Q.

4 YURI LIMA AND CARLOS GUSTAVO MOREIRA
In this case, proj θ ((K 1 ×K 2 ) ρ ′) ⊆ proj θ ((K 1 ×K 2 ) ρ ) for any θ.
A remarkable property of regular Cantor sets of class C 1+α , α > 0, is bounded
distortion.
Lemma 2.3. Let (K,ψ) be a regular Cantor set of class C 1+α , α > 0, and
{I 1 ,...,I r } a Markov partition. Given δ > 0, there exists a constant C(δ) > 0,
decreasing on δ, with the following property: if x,y ∈ K satisfy
then
(i) |ψ n (x)−ψ n (y)| < δ;
(ii) The interval [ψ i (x),ψ i (y)] is contained in I 1 ∪···∪I r , for i = 0,...,n−1,
In addition, C(δ) → 0 as δ → 0.
e −C(δ) ≤ |(ψn ) ′ (x)|
|(ψ n ) ′ (y)| ≤ eC(δ) .
A direct consequence of bounded distortion is the required regularity of K, contained
in the next result.
Lemma 2.4. Let K be a regular Cantor set of class C 1+α , α > 0, and let d =
HD(K). Then 0 < m d (K) < +∞. Moreover, there is c > 0 such that, for any
x ∈ K and 0 ≤ r ≤ 1,
c −1 ·r d ≤ m d (K ∩B r (x)) ≤ c·r d .
ThesamehappensforproductsK 1 ×K 2 ofCantorsets(withoutlossofgenerality,
considered with the box norm).
Lemma 2.5. Let K 1 ,K 2 be regular Cantor sets of class C 1+α , α > 0, and let
d = HD(K 1 )+HD(K 2 ). Then 0 < m d (K 1 ×K 2 ) < +∞. Moreover, there is c 1 > 0
such that, for any x ∈ K 1 ×K 2 and 0 ≤ r ≤ 1,
c 1
−1 ·r d ≤ m d ((K 1 ×K 2 )∩B r (x)) ≤ c 1 ·r d .
See chapter 4 of [9] for the proofs of these lemmas. In particular, if Q ∈ (K 1 ×
K 2 ) ρ , there is x ∈ (K 1 ∪K 2 )∩Q such that B λ −1 ρ(x) ⊆ Q ⊆ B λρ (x) and so
(
c1 λ d) −1
·ρ d ≤ m d ((K 1 ×K 2 )∩Q) ≤ c 1 λ d ·ρ d .
Changing c 1 by c 1 λ d , we may also assume that
c 1
−1 ·ρ d ≤ m d ((K 1 ×K 2 )∩Q) ≤ c 1 ·ρ d ,
which allows us to obtain estimates on the cardinality of ρ-decompositions.
Lemma 2.6. Let K 1 ,K 2 be regular Cantor sets of class C 1+α , α > 0, and let
d = HD(K 1 ) +HD(K 2 ). Then there is c 2 > 0 such that, for any ρ-decomposition
(K 1 ×K 2 ) ρ , x ∈ K 1 ×K 2 and 0 ≤ r ≤ 1,
( ) r
#{Q ∈ (K 1 ×K 2 ) ρ ;Q ⊆ B r (x)} ≤ c 2 ·
d·
ρ
In addition, c 2
−1 ·ρ −d ≤ #(K 1 ×K 2 ) ρ ≤ c 2 ·ρ −d .

MARSTRAND’S THEOREM FOR PRODUCTS OF CANTOR SETS 5
Proof. We have
and then
c 1 ·r d
≥ m d ((K 1 ×K 2 )∩B r (x))
∑
≥ m d ((K 1 ×K 2 )∩Q)
≥
Q⊆B r(x)
∑
Q⊆B r(x)
c 1
−1 ·ρ d
= #{Q ∈ (K 1 ×K 2 ) ρ ;Q ⊆ B r (x)}·c 1
−1 ·ρ d
( ) r
#{Q ∈ (K 1 ×K 2 ) ρ ;Q ⊆ B r (x)} ≤ c 12 ·
d·
ρ
On the other hand,
∑
∑
m d (K 1 ×K 2 ) = m d ((K 1 ×K 2 )∩Q) ≤ c 1 ·ρ d ,
Q∈(K 1×K 2) ρ Q∈(K 1×K 2) ρ
implying that
−1 #(K 1 ×K 2 ) ρ ≥ c 1 ·m d (K 1 ×K 2 )·ρ −d .
Taking c 2 = max{c 2 1 , c 1 /m d (K 1 ×K 2 )}, we conclude the proof.
□
3. Proof of Theorem 1.1
Given rectangles Q and ˜Q, let
{
}
Θ Q, ˜Q
= θ ∈ [−π/2,π/2];proj θ (Q)∩proj θ (˜Q) ≠ ∅ .
Lemma 3.1. If Q, ˜Q ∈ (K 1 ×K 2 ) ρ and x ∈ (K 1 ×K 2 )∩Q,˜x ∈ (K 1 ×K 2 ) ∩ ˜Q,
then
( )
ρ
m Θ Q, ˜Q
≤ 2πλ·
d(x,˜x) ·
Proof. Consider the figure.
x
θ |θ −ϕ 0|
˜x
L θ
proj θ (˜x)
θ
proj θ (x)
Since proj θ (Q) has diameter at most λρ, d(proj θ (x),proj θ (˜x)) ≤ 2λρ and then, by
elementary geometry,
sin(|θ −ϕ 0 |) = d(proj θ (x),proj θ (˜x))
d(x,˜x)
ρ
≤ 2λ·
d(x,˜x)
ρ
=⇒ |θ −ϕ 0 | ≤ πλ·
d(x,˜x) ,

6 YURI LIMA AND CARLOS GUSTAVO MOREIRA
because sin −1 y ≤ πy/2. As ϕ 0 is fixed, the lemma is proved.
□
Wepointoutthat, althoughingenuous,Lemma3.1expressesthecrucialproperty
of transversality that makes the proof work, and all results related to Marstrand’s
theorem use a similar idea in one way or another. See [11] where this tranversality
condition is also exploited.
Fixed a ρ-decomposition (K 1 ×K 2 ) ρ , let
{
N (K1×K 2) ρ
(θ) = # (Q, ˜Q)
}
∈ (K 1 ×K 2 ) ρ ×(K 1 ×K 2 ) ρ ;proj θ (Q)∩proj θ (˜Q) ≠ ∅
for each θ ∈ [−π/2,π/2] and
E((K 1 ×K 2 ) ρ ) =
∫ π/2
−π/2
N (K1×K 2) ρ
(θ)dθ.
Proposition 3.2. Let K 1 ,K 2 be regular Cantor sets of class C 1+α , α > 0, and let
d = HD(K 1 ) +HD(K 2 ). Then there is c 3 > 0 such that, for any ρ-decomposition
(K 1 ×K 2 ) ρ ,
E((K 1 ×K 2 ) ρ ) ≤ c 3 ·ρ 1−2d .
Proof. Let s 0 = ⌈ log 2 ρ −1⌉ and choose, for each Q ∈ (K 1 × K 2 ) ρ , a point x ∈
(K 1 ×K 2 )∩Q. By a double counting and using Lemmas 2.6 and 3.1, we have
∑ ( )
E((K 1 ×K 2 ) ρ ) = Θ Q, ˜Q
=
Q, ˜Q∈(K 1×K 2) ρ
m
s 0
∑
s=1
∑
m
Q, ˜Q∈(K 1 ×K 2 )ρ
2 −s 1, c 3 = 2 d+1 πλc 22 ·∑s≥1 2s(1−d) < +∞ satisfies the required inequality.
□
This implies that, for each ε > 0, the upper bound
N (K1×K 2) ρ
(θ) ≤ c 3 ·ρ 1−2d
ε
(3.1)
holds for every θ except for a set of measure at most ε. Letting c 4 = c 2
−2 · c 3 −1 ,
we will show that
m(proj θ ((K 1 ×K 2 ) ρ )) ≥ c 4 ·ε (3.2)
for every θ satisfying (3.1). For this, divide [−2,2] ⊆ L θ in ⌊4/ρ⌋ intervals J ρ 1 ,...,
J ρ ⌊4/ρ⌋
of equal lenght (at least ρ) and define
s ρ,i = #{Q ∈ (K 1 ×K 2 ) ρ ; proj θ (x) ∈ J ρ i }, i = 1,...,⌊4/ρ⌋.

MARSTRAND’S THEOREM FOR PRODUCTS OF CANTOR SETS 7
Then ∑ ⌊4/ρ⌋
i=1
s ρ,i = #(K 1 ×K 2 ) ρ and
⌊4/ρ⌋
∑
i=1
s ρ,i 2 ≤ N (K1×K 2) ρ
(θ) ≤ c 3 ·ρ 1−2d ·ε −1 .
Let S ρ = {1 ≤ i ≤ ⌊4/ρ⌋;s ρ,i > 0}. By Cauchy-Schwarz inequality,
⎛ ⎞
⎝ ∑
2
s ρ,i
⎠
i∈S ρ
#S ρ ≥ ∑ ≥ c 2 −2 ·ρ −2d
c 3 ·ρ 1−2d ·ε −1 = c 4 ·ε
·
ρ
i∈S ρ
s ρ,i
2
For each i ∈ S ρ , the interval J ρ i is contained in proj θ ((K 1 ×K 2 ) ρ ) and then
which proves (3.2).
m(proj θ ((K 1 ×K 2 ) ρ )) ≥ c 4 ·ε,
Proof of Theorem 1.1. Fix a decreasing sequence
(K 1 ×K 2 ) ρ1 ≻ (K 1 ×K 2 ) ρ2 ≻ ··· (3.3)
of decompositions such that ρ n → 0 and, for each ε > 0, consider the sets
G n ε = { θ ∈ [−π/2,π/2]; N (K1×K 2) ρn
(θ) ≤ c 3 ·ρ n
1−2d ·ε −1} , n ≥ 1.
Then m([−π/2,π/2]\G n ε ) ≤ ε, and the same holds for the set
G ε = ⋂ ∞⋃
G l ε .
n≥1
If θ ∈ G ε , then
l=n
m(proj θ ((K 1 ×K 2 ) ρn )) ≥ c 4 ·ε, for infinitely many n,
which implies that m(proj θ (K 1 ×K 2 )) ≥ c 4 · ε. Finally, the set G = ∪ n≥1 G 1/n
satisfies m([−π/2,π/2]\G) = 0 and m(proj θ (K 1 ×K 2 )) > 0, for any θ ∈ G. □
4. Proof of Theorem 1.2
Given any X ⊂ K 1 × K 2 , let (X) ρ be the restriction of the ρ-decomposition
(K 1 × K 2 ) ρ to those rectangles which intersect X. As done in Section 3, we’ll
obtain estimates on the cardinality of (X) ρ . Being a subset of K 1 ×K 2 , the upper
estimates from Lemma 2.6 also hold for X. The lower estimate is given by
Lemma 4.1. Let X be a subset of K 1 ×K 2 such that m d (X) > 0. Then there is
c 6 = c 6 (X) > 0 such that, for any ρ-decomposition (K 1 ×K 2 ) ρ and 0 ≤ r ≤ 1,
c 6 ·ρ −d ≤ #(X) ρ ≤ c 2 ·ρ −d .
Proof. As m d (X) < +∞, there exists c 5 = c 5 (X) > 0 (see Theorem 5.6 of [1]) such
that
m d (X ∩B r (x)) ≤ c 5 ·r d , for all x ∈ X and 0 ≤ r ≤ 1,
and then
m d (X) = ∑
m d (X ∩Q) ≤ ∑
c 5 ·(λρ) d = ( c 5 ·λ d)·ρ d ·#(X) ρ .
Q∈(X) ρ Q∈(X) ρ
Just take c 6 = c 5
−1 ·λ −d ·m d (X).
□

8 YURI LIMA AND CARLOS GUSTAVO MOREIRA
Proposition 4.2. The measure µ θ = (proj θ ) ∗ (m d | K1×K 2
) is absolutely continuous
with respect to m, for m-almost every θ ∈ R.
Proof. Note that the implication
X ⊂ K 1 ×K 2 , m d (X) > 0 =⇒ m(proj θ (X)) > 0 (4.1)
is sufficient for the required absolute continuity. In fact, if Y ⊂ L θ satisfies m(Y) =
0, then
µ θ (Y) = m d (X) = 0,
whereX = proj θ −1 (Y). Otherwise,by(4.1)wewouldhavem(Y) = m(proj θ (X)) >
0, contradicting the assumption.
We prove that (4.1) holds for every θ ∈ G, where G is the set defined in the
proof of Theorem 1.1. The argument is the same made after Proposition 3.2: as,
by the previous lemma, #(X) ρ has lower and upper estimates depending only on
X and ρ, we obtain that
m(proj θ ((X) ρn )) ≥ c 3
−1 ·c 62 ·ε, for infinitely many n,
and then m(proj θ (X)) > 0.
Let χ θ = dµ θ /dm. In principle, this is a L 1 function. We prove that it is a L 2
function, for every θ ∈ G.
Proof of Theorem 2. Let θ ∈ G 1/m , for some m ∈ N. Then
N (K1×K 2) ρn
(θ) ≤ c 3 ·ρ n
1−2d ·m, for infinitely many n. (4.2)
For each of these n, consider the partition P n = {J ρn
1 ,...,Jρn ⌊4/ρ n⌋ } of [−2,2] ⊂ L θ
into intervals of equal length and let χ θ,n be the expectation of χ θ with respect
to P n . As ρ n → 0, the sequence of functions (χ θ,n ) n∈N converges pointwise to χ θ .
By Fatou’s Lemma, we’re done if we prove that each χ θ,n is L 2 and its L 2 -norm
‖χ θ,n ‖ 2
is bounded above by a constant independent of n.
By definition,
and then
µ θ (J ρn
i
) = m d
(
(projθ ) −1 (J ρn
i
) ) ≤ s ρn,i ·c 1 ·ρ n d , i = 1,2,...,⌊4/ρ n ⌋,
implying that
i )
|J ρn
i |
χ θ,n (x) = µ θ(J ρn
‖χ θ,n ‖ 2 2
=
=
≤
∫
≤ c 1 ·s ρn,i ·ρ n
d
|J ρn
i |
L θ
|χ θ,n | 2 dm
⌊4/ρ
∑ n⌋
i=1
⌊4/ρ
∑ n⌋
i=1
∫
J ρn
i
|J ρn
i |·
|χ θ,n | 2 dm
⌊4/ρ
∑ n⌋
2d−1 ≤ c 12 ·ρ n ·
, ∀x ∈ J ρn
i ,
(
c1 ·s ρn,i ·ρ n
d
i=1
|J ρn
i |
s ρn,i 2
≤ c 12 ·ρ n
2d−1 ·N (K1×K 2) ρn
(θ).
) 2
□

MARSTRAND’S THEOREM FOR PRODUCTS OF CANTOR SETS 9
In view of (4.2), this last expression is bounded above by
(
c12 ·ρ n
2d−1 )·(c 3 ·ρ n
1−2d ·m ) = c 12 ·c 3 ·m,
which is a constant independent of n.
□
5. Concluding remarks
The proofs of Theorems 1.1 and 1.2 work not just for the case of products of
regular Cantor sets, but in greater generality, whenever K ⊂ R 2 is a Borel set for
which there is a constant c > 0 such that, for any x ∈ K and 0 ≤ r ≤ 1,
c −1 ·r d ≤ m d (K ∩B r (x)) ≤ c·r d ,
since this alone implies the existence of ρ-decompositions for K.
The good feature of the proof is that the discretization idea may be applied
to other contexts. For example, we prove in [3] a Marstrand type theorem in an
arithmetical context.
Acknowledgments
The authors are thankful to IMPA for the excellent ambient during the preparation
of this manuscript. The authors are also grateful to Carlos Matheus for
carefully reading the preliminary version of this work and the anonymous referee
for many useful and detailed recommendations. This work was financially supported
by CNPq-Brazil and Faperj-Brazil.
References
1. K. Falconer, The geometry of fractal sets, Cambridge Tracts in Mathematics, Cambridge
(1986).
2. R. Kaufman, On Hausdorff dimension of projections, Mathematika 15 (1968), 153–155.
3. Y. Lima and C.G. Moreira, A Marstrand theorem for subsets of integers, available at
http://arxiv.org/abs/1011.0672.
4. J.M. Marstrand, Some fundamental geometrical properties of plane sets of fractional dimensions,
Proceedings of the London Mathematical Society 3 (1954), vol. 4, 257–302.
5. P. Mattila, Hausdorff dimension, projections, and the Fourier transform, Publ. Mat. 48
(2004), no. 1, 3–48.
6. C.G. Moreira, A dimension formula for arithmetic sums of regular Cantor sets, to appear.
7. C.G. Moreira and J.C. Yoccoz, Stable Intersections of Cantor Sets with Large Hausdorff
Dimension, Annals of Mathematics 154 (2001), 45–96.
8. C.G. Moreira and J.C. Yoccoz, Tangences homoclines stables pour des ensembles hyperboliques
de grande dimension fractale, Annales scientifiques de l’ENS 43, fascicule 1 (2010).
9. J. Palis and F. Takens, Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations,
Cambridge studies in advanced mathematics, Cambridge (1993).
10. J. Palis and J.C. Yoccoz, On the Arithmetic Sum of Regular Cantor Sets, Annales de
l’Inst. Henri Poincar, Analyse Non Lineaire 14 (1997), 439–456.
11. M. Rams, Exceptional parameters for iterated function systems with overlaps, Period. Math.
Hungar. 37 (1998), no. 1-3, 111–119.
12. M. Rams, Packing dimension estimation for exceptional parameters, Israel J. Math. 130
(2002), 125–144.
Instituto Nacional de Matemática Pura e Aplicada, Estrada Dona Castorina 110,
22460-320, Rio de Janeiro, Brasil.
E-mail address: yurilima@impa.br
Instituto Nacional de Matemática Pura e Aplicada, Estrada Dona Castorina 110,
22460-320, Rio de Janeiro, Brasil.
E-mail address: gugu@impa.br