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

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 .