A generalization of Marstrand's theorem for projections of cartesian products

ScienceDirect

Ann. I. H. Poincaré – AN 32 (2015) 833–840

www.elsevier.com/locate/anihpc

A generalization of Marstrand’s theorem for projections of cartesian products

Jorge Erick López, Carlos Gustavo Moreira

IMPA, Estrada D. Castorina 110, 22460-320, Rio de Janeiro, RJ, Brazil

Received 18 September 2013; received in revised form 11 March 2014; accepted 1 April 2014 Available online 18 April 2014

Abstract

We prove the following variant of Marstrand’s theorem about projections of cartesian products of sets:

LetK₁, . . . , K_nbe Borel subsets ofR^m1, . . . ,R^mnrespectively, andπ:R^m1×. . .×R^mn→R^kbe a surjective linear map. We set

m:=min i∈I

dim_H(K_i)+dimπ i∈I^c

R^mi

, I⊂ {1, . . . , n}, I= ∅

.

Consider the spaceΛ_m= {(t, O), t∈R, O∈SO(m)}with the natural measure and setΛ=Λ_m1×. . .×Λ_mn. For every λ=(t₁, O₁, . . . , t_n, O_n)∈Λand everyx=(x¹, . . . , xⁿ)∈R^m1×. . .×R^mnwe defineπ_λ(x)=π(t₁O₁x¹, . . . , t_nO_nxⁿ). Then we have

Theorem.

(i) Ifm> k, thenπ_λ(K₁×. . .×K_n)has positivek-dimensional Lebesgue measure for almost everyλ∈Λ.

(ii) Ifmkanddim_H(K₁×. . .×Kn)=dim_H(K₁)+. . .+dim_H(Kn), thendim_H(π_λ(K₁×. . .×Kn))=mfor almost every λ∈Λ.

©2014 Published by Elsevier Masson SAS.

Keywords:Fractal geometry; Hausdorff dimensions; Potential theory; Fourier transform; Dynamical systems

✩ The first author was supported by the Balzan Research Project of J. Palis and by INCTMat. The second author was partially supported by the Balzan Research Project of J. Palis and by CNPq.

* Corresponding author.

E-mail addresses:[email protected](J.E. López),[email protected](C.G. Moreira).

http://dx.doi.org/10.1016/j.anihpc.2014.04.002 0294-1449/©2014 Published by Elsevier Masson SAS.

1. Introduction

Let us denote by dimH(X)the Hausdorff dimension of the setX. Fornandkintegers with 0< k < n,Π_n.kdenotes the space of orthogonal projections fromRⁿtok-dimensional subspaces ofRⁿ, with natural measure. A fundamental result in dimensions of projections is the following theorem:

Theorem (Marstrand–Kaufman–Mattila). LetE⊂Rⁿa Borel set. Then:

(i) IfdimH(E) > k, thenπ(E)has positivek-dimensional Lebesgue measure for almost everyπ∈Π_n.k. (ii) IfdimH(E)k, thendimH(π(E))=dimH(E)for almost everyπ∈Π_n.k.

This theorem was first proven for planar sets by Marstrand[3]. Marstrand’s proof used geometric methods. Later, Kaufman[2]gave an alternative proof of the same result applying potential-theoretic methods. Finally, Mattila[4]

generalized it to higher dimensions; his proof combines Marstrand and Kaufman methods.

There are other variants of Marstrand–Mattila’s theorem that were unified in a more general result due to Peres and Schlag [7]. These authors studied general smooth families of projections, using some methods from harmonic analysis. The crucial characteristic that is common to all families of projections considered in Peres–Schlag’s result is a transversality property (see[7, Definition 7.2]).

We are interested in Marstrand’s projection result that actually is outside of Peres–Schlag’s scheme (the families of projections considered here, in general, are not transversal). This result was motivated by the problem of understanding the behavior of projections of cartesian products of sets, by a fixed projection map.

LetK₁, . . . , K_n be Borel subsets ofR^m1, . . . ,R^mn respectively, andπ:R^m1 ×. . .×R^mn→R^k be a linear map.

Then dimH

π(K₁×. . .×K_n) min

i∈I

dimH(K_i)+dimπ

i∈I^c

R^mi

, I⊂ {1, . . . , n}

, (1.1)

with the conventions

i∈∅dimH(K_i)=0, dim∅ =0.

Consider the space Λ_m= {(t, O), t ∈R, O ∈SO(m)} with the natural measure and set Λ=Λ_m1 ×. . .× Λ_mn. For every x =(x¹, . . . , xⁿ)∈R^m1 ×. . .×R^mn and every λ=(t₁, O₁, . . . , t_n, O_n)∈Λ we defineπ_λ(x)= π(t1O1x¹, . . . , t_nO_nxⁿ). Suppose thatπ is surjective and set

m:=min

i∈I

dimH(K_i)+dimπ

i∈I^c

R^mi

, I⊂ {1, . . . , n}, I= ∅

.

Then we have Theorem 1.1.

(i) Ifm> k, thenπ_λ(K₁×. . .×K_n)has positivek-dimensional Lebesgue measure for almost everyλ∈Λ.

(ii) Ifmk anddimH(K₁×. . .×K_n)=dimH(K₁)+. . .+dimH(K_n), thendimH(π_λ(K₁×. . .×K_n))=mfor almost everyλ∈Λ.

We recover Marstrand–Mattila’s theorem considering the cartesian product of only one set.

Theorem 2.3is a fundamental tool in our forthcoming work which generalizes the result of Moreira and Yoccoz[6]

about stable intersections of two regular Cantor sets for projections of cartesian products of several regular Cantor sets. We prove the following result: for any given surjective linear mapπ:Rⁿ→R^k, typically for regular Cantor sets on the real lineK₁, . . . , K_n withm> k, the setπ(K₁×. . .×K_n)persistently contains non-empty open sets ofR^k. Such a result in particular implies an analogous result for simultaneous stable intersections of several regular Cantor sets on the real line.

In another forthcoming work, in collaboration with Pablo Shmerkin, we use the results of this paper combined with the techniques in[1]in order to obtain exact formulas for the Hausdorff dimensions of projections of cartesian products of (real or complex) regular Cantor sets under explicit irrationality conditions.

2. Statement the main results

Letμbe a finite Borel measure onR^m. Thes-energyofμis I_s(μ)=

¨ dμ(x)dμ(y)

|x−y|^s .

We know (see[5, Theorem 8.9(3)]) that for a Borel setK⊂R^m dimH(K)=sup

s∈R, there is a compactly supported measureμonK with 0< μ

R^m <∞andI_s(μ) <∞

. (2.1)

TheFourier transformofμis denoted byμˆ and defined as ˆ

μ(ξ )= ˆ

R^m

e^−iξ·xdμ(x).

It is well-known that ifμˆ ∈L²(R^m), thenμis absolutely continuous withL²-density. Energy and Fourier transform are related as follows (see[5, Lemma 12.12])

Is(μ)=(2π )^−mc(s, m) ˆ

|ξ|^s−mμ(ξ )ˆ ²dξ, for 0< s < mandμwith compact support.

We summarize the above observations as the following result. LetF ⊂R^k a Borel set supporting a probability measureνwith´

|ξ|^s−k|ˆν(ξ )|²dξ <∞. Ifsk, thenF has positivek-dimensional Lebesgue measure. Otherwise, if 0< s < k, then dimH(F )s.

Letπ:R^m1×. . .×R^mn→R^k be a linear map. For eachI⊂ {1, . . . , n}, letP_I:R^m1×. . .×R^mn→R^m1×. . .× R^mnthe orthogonal projection onto the subspace

i∈IR^mi, whereR^miis as a canonical subspace ofR^m1×. . .×R^mn. Thenπ=π◦P_I+π◦P_I^cso, forK₁, . . . , K_nBorel subsets ofR^m1, . . . ,R^mnrespectively we have

dimH

π(K₁×. . .×K_n) dimH

π P_I(K₁×. . .×K_n)×π P_I^c(K₁×. . .×K_n) dimH

π PI(K1×. . .×Kn)×π

i∈I^c

R^mi

i∈I

dimH(K_i)+dimπ

i∈I^c

R^mi

.

(In the last inequality, we assume that dimH(K1×. . .×Kn)=dimH(K1)+. . .+dimH(Kn).) This proves the in- equality(1.1)and also motivates us to define:

Definition 2.1.Forπ:R^m1×. . .×R^mn→R^ka surjective linear map andd₁, . . . , d_nnonnegative real numbers, we definem=m(π, d₁, . . . , d_n)as

m=min

i∈I

di+dimπ

i∈I^c

R^mi

, I⊂ {1, . . . , n}, I= ∅

.

Remark 2.2.If in addition d_i m_i (which holds for dimensions of subsets of R^mi), then, for the open and total measure family of linear mapsπwith the following transversality property:

dimπ

i∈I

R^mi

=min

k,dim

i∈I

R^mi

, for allI⊂ {1, . . . , n},

the equivalencem(π, d₁, . . . , d_n) > k⇔d₁+. . .+d_n> kholds. However, in general we must check more than one of the 2ⁿ−1 conditions appearing in the definition ofm.

Consider the space Λ_m= {(t, O), t ∈R, O∈SO(m)}, with the product measureL¹×Θ^m, whereL¹denotes the one dimensional Lebesgue measure andΘ^mdenotes the bi-invariant Haar probability measure onSO(m). Notice that the set C(m)= {t O, t∈R, O∈SO(m)}represents essentially the family of linear conformal maps on R^m. C(2)= {

a−b b a

, a, b∈R}, which can be viewed as the set of multiplications by a complex number.

We setΛ=Λm₁×. . .×Λm_n. For everyx=(x¹, . . . , xⁿ)∈R^m1×. . .×R^mn, and everyλ=(t1, O1, . . . , tn, On)∈ Λwe defineπλ(x)=π(t1O1x¹, . . . , tnOnxⁿ). Also, given any finite measureμonR^m1×. . .×R^mn, letνλ=(πλ)_∗μ.

We also define Id₁,...,d_n(μ)=

¨ dμ(x)dμ(y)

|x¹−y¹|^d1. . .|xⁿ−yⁿ|^dn. Our main result is now the following:

Theorem 2.3.Letπandd₁, . . . , d_nbe as inDefinition 2.1withm=m(π, d₁, . . . , d_n)=0,1, . . . , k−1. Then, there existd₁d₁, . . . , d_n d_nsuch that for every Borel measureμonR^m1×. . .×R^mn we have

ˆ

Λ

ˆ

R^k

|ξ|^m−kνλ(ξ )²ρ(λ)dξ dλCmI_d

1,...,d_n(μ),

where ρ(λ)= |t₁|^m1−1. . .|t_n|^mn−1e^{−12(|t1|2+...+|tn|2)} and Cm>0 is some constant depending only on π, n, k, m₁, . . . , m_nandm.

In the proof ofTheorem 2.3the key tool will be the following combinatorial lemma.

Lemma 2.4(Weights lemma). Lets, d₁, . . . , d_n0andV₁, . . . , V_nbe vector subspaces of a fixed finite dimensional vector space satisfying the following2ⁿconditions

i∈I

d_i+dim

i∈I^c

V_i

s, for everyI⊂ {1, . . . , n} (with the conventions

i∈∅d_i=0,dim∅ =0).

Fix a generating set {v₁ⁱ, . . . , v_mⁱ

i} of V_i for each i∈ {1, . . . , n}. Consider the family J of all possible J = (_J1, . . . ,Jn),Ji ⊂ {v₁ⁱ, . . . , v_mⁱ

i}such thatJ1∪. . .∪^Jnis a linearly independent system with dimension greater than or equal tos. Define

J=

(J, i)∈J× {1, . . . , n}, J (i) :=(#J1, . . . ,#Jn)+

s−(#J1+. . .+#Jn)e_i0 ,

wheree₁, . . . , e_nis the canonical basis ofRⁿandmeans that the inequality is coordinate to coordinate.

Then, there exist non-negative real numbers(α_(J,i))_(J,i)∈Jwith sum equal to1such that

(J,i)∈J

α(J,i)J (i) d:=(d1, . . . , dn).

Proof of Theorem 1.1. The theorem follows immediately from Theorem 2.3applied to μ=μ₁×. . .×μ_n for suitable measuresμ_i compactly supported inK_i coming from Eq.(2.1). Indeed, this is so because in the part (i) the condition dimH(K₁) >0, . . . ,dimH(K_n) >0 follows from the hypotheses, and in the part (ii) we may assume the same condition by reduction to some cartesian product if necessary. 2

Remark 2.5. We can derive the part (ii) ofTheorem 1.1from the part (i). Assume dimH(K_i) >0. Letk<m k+1kand consider anyk< s <m, and setΛ^s = {λ∈Λ,dimH(π_λ(K1×. . .×K_n)) < s}. The idea is to add another factor to the cartesian product: Let m0:=k−k and considerK0a sufficiently regular subset ofR^m0 with dimH(K₀)=k−s, andπ˜:R^m0×R^m1×. . .×R^mn→R^k with

˜

π◦P_Iⁿ=π, whereIⁿ= {1, . . . , n}, dimπ˜

i∈I∪{0}

R^mi

=min

k, m₀+dimπ

i∈I

R^mi

, for allI⊂ {1, . . . , n}. In particularπ˜ is surjective. Notice that

i∈I

dimH(Ki)+dimπ˜

i∈I^c

R^mi

> k, for allI⊂ {0,1, . . . , n}, I= ∅,

and also that dimH(π˜_(λ0,λ)(K₀×K₁×. . .×K_n)) < kfor all(λ₀, λ)∈Λ_m0×Λ^s. ApplyingTheorem 1.1(i) in this new setting, we conclude thatΛ^sis a zero measure subset ofΛ.

Remark 2.6.Theorem 2.3, when combined with Proposition 7.5 of[7], also gives us a result on exceptional sets:

In the setting ofTheorem 1.1, part (i), we have dimH

λ∈Λ, ti=0 ifmi>1,L^k

πλ(K1×. . .×Kn) =0 l+k−m, wherel=dimΛm₁×. . .×Λm_n=n+n

i=1mi(mi−1)/2.

3. Proof of the main results

Proof of Theorem 2.3 assuming Lemma 2.4. Notice that ν_λ(ξ )²=

¨

e^{iξ·πλ(y−x)}dμ(x)dμ(y),

=

¨

e^{iπTξ·(t1O1(y1−x1),...,tnOn(yn−xn))}dμ(x)dμ(y), and that, for allz∈R^m,η∈R^m,

ˆ

R

ˆ

SO(m)

e^{iη·t Oz}|t|^m−1e^−12|t|2dΘ^mdt= ˆ

R

ˆ

S^m−1

e^{i|z|η·t θ}|t|^m−1e^−12|t|2dσ^m−1dt

=2 ˆ

R^m

e^i|z|η·xe^−12|x|2dx

=2π^m2e^{−12(|z||η|)2},

whereσ^m−1denotes the normalized Lebesgue measure onS^m−1. Therefore by Fubini’s theorem ˆ

Λ

ˆ

R^k

|ξ|^m−kν_λ(ξ )²ρ(λ)dξ dλ= lim

a→∞

ˆ

|ξ|a

ˆ

Λ

|ξ|^m−kν_λ(ξ )²ρ(λ)dλdξ

=c lim

a→∞

¨ ˆ

|ξ|a

|ξ|^m−ke^{−12|Dx,y(ξ )|2}dξ

dμ(x)dμ(y)

=c

¨ ˆ

R^k

|ξ|^m−ke^{−12|Dx,y(ξ )|2}dξ

dμ(x)dμ(y),

whereD_x,y =(D¹(|y¹−x¹|), . . . , Dⁿ(|yⁿ−xⁿ|))◦π^T, and Dⁱ(t):R^mi →R^mi is the diagonal transformation, Dⁱ(t)=t.Id, fort∈R.

We fixx, yassuming thatyⁱ−xⁱ=0 for alli=1, . . . , n. We estimate´

R^k|ξ|^m−ke^{−12|Dx,y(ξ )|2}dξseparately, when mkandm< k. In both cases we applyLemma 2.4forV_i=π(R^mi), takingv_jⁱ =π(eⁱ_j), whereeⁱ_j,j=1, . . . , miis the canonical basis ofR^mias subspace ofR^m1×. . .×R^mn.

We use the notationz^I =zⁱ₁¹. . . zⁱ_nⁿ if z=(z₁, . . . , z_n)∈Rⁿ₊ andI =(i₁, . . . , i_n)∈Zⁿ, for z=(|y¹−x¹|, . . . ,

|yⁿ−xⁿ|).

Suppose mk. Let i₀ be such thatz_i0 z_i for all i=1, . . . , n. Notice thatm(π, d−(m−k)e_i0)k and in particular d−(m−k)e_i0 0. We applyLemma 2.4tod−(m−k)e_i0 ands=k. For eachJ ∈J, just looking for the sums in ¹₂|Dx,y(ξ )|²related toJ and using the change of variables formula to an appropriate linear isomorphism ofR^k, we have

ˆ

R^k

|ξ|^m−ke^{−12|Dx,y(ξ )|2}dξcz^k_i^−m

0 z^−J ˆ

R^k

|η|^m−ke^−12|η|2dη,

for some constantc>0 depending only onπ,mandk, whereJ:=(#J1, . . . ,#Jn). Therefore ˆ

R^k

|ξ|^m−ke^{−12|Dx,y(ξ )|2}dξcz^k_i^−m

0

J∈J

z^−αJJ=cz^{−(JαJJ+(m−k)ei0)}=:cz^−d.

Supposem< k. We applyLemma 2.4todands=m. Let(J, i)∈J. We definek:=#J1+. . .+#Jnandk_i:=#Ji, thenm< kandk_i> k−m. Similary to the previous case, notice that

ˆ

R^k

|ξ|^m−ke^{−12|Dx,y(ξ )|2}dξcz˜ ^−J ˆ

R^ki

ˆ

R^k−k

η/z_i+η ^m−ke^−12|η|2dηdη, for some constantc >˜ 0 depending only onπ,m, k, k, k_i. We affirm that

ˆ

R^ki

ˆ

R^k−k

η/z_i+η ^m−ke^−12|η|2dηdηc˜z_i^k−m,

for some constantc˜>0 depending only onm, k, k, k_i. Ifk=kthe affirmation is true, sincem−k>−k_i. Ifk< k, applying polar coordinates inR^k−k we have

ˆ

R^ki

ˆ

R^k−k

η/z_i+η ^m−ke^−12|η|2dηdηC ˆ

R^ki

ˆ

R+

η/z_i+r ^m−k−1e^−12|η|2drdη

=C

k−m ⁻¹ ˆ

R^ki

η/zi m−k

e^−12|η|2dη.

Then´

R^k|ξ|^m−ke^{−12|Dx,y(ξ )|2}dξc˜z^{−J (i)} , and therefore ˆ

R^k

|ξ|^m−ke^{−12|Dx,y(ξ )|2}dξc˜

(J,i)∈J

z^{−α(J,i)J (i)} = ˜cz⁻

(J,i)∈Jα_(J,i)J (i) =: ˜cz^−d. 2

Proof of Lemma 2.4.

Claim.The vertices of the polyhedron

P =

(d₁, . . . , d_n)∈Rⁿ, d₁0, . . . , dn0,

i∈I

d_i+dim

i∈I^c

V_i

s, for allI⊂ {1, . . . , n}

have all the formJ (i) for some(J, i)∈J.

P ⊂Rⁿ₊, thereforeP is a pointed polyhedron (i.e. it does not contain any non-trivial affine subspace). We pro- ceed by induction onn. Forn=1 it is trivial. Letx=(x₁, . . . , x_n)any vertex of the polyhedron. Then, there aren independent inequalities from the definition ofP that become equality atx(see[8, p. 104]).

Ifx_n=0, notice thatx=(x₁, . . . , x_n−1)is now a vertex of the polyhedron P=

(d1, . . . , dn−1)∈Rⁿ⁻¹, d10, . . . , dn−10,

i∈I

di+dim

i∈I^c

Vi

s, for allI ⊂ {1, . . . , n−1}

(i.e. x ∈P and x satisfies n −1 independent equalities). By induction hypothesis, there exist some J = (J

1, . . . ,J

n−1)∈Jandi∈ {1, . . . , n−1}such thatx=J(i). Then,J =(J 1, . . . ,J

n−1,∅)∈Jandi=iare such thatx=J (i).

Supposex₁=0, . . . , xn=0. By simplicity, we denote

i∈IV_i byV_I. Consider I=

I⊂ {1, . . . , n}, I= ∅,

i∈I

xi+dimVI^c=s

.

By the assumption onx, there areI1, . . . , In∈I such that the associated 0,1 row vectors I1, . . . ,In defining the equalities, are independent.

IfI, J∈I, then

dimV_I^c+dimV_J^c=2s−

i∈I

x_i−

i∈J

x_i

=2s−

i∈I∪J

x_i−

i∈I∩J

x_i dimV_Ic∩J^c+dimV_Ic∪J^c

dim(VI^c∩V_J^c)+dim(VI^c+V_J^c)

=dimV_I^c+dimV_J^c,

therefore,I∪J∈IandI∩J∈I. LetI0∈I, I0= ∅a minimal element by inclusion. Then, for anyJ∈I, we have I0⊂J or I0∩J= ∅.

This means the invertible matrix of rowsI₁, . . . ,I_nhas #I0identical columns, and therefore #I0=1, sayI₀= {n}, or, equivalently,x_n=s−dim(V1+. . .+V_n−1).

Notice that nowx˜=(x₁, . . . , x_n−1)is a vertex of the polyhedron P=

(d₁, . . . , d_n−1)∈Rⁿ⁻¹, d₁0, . . . , dn−10,

i∈I

d_i+dim

i∈I^c

V_i

dim(V1+. . .+V_n−1),for allI⊂ {1, . . . , n−1}

.

By induction hypothesis, there exist some appropriateJ=(˜J1, . . . ,˜^Jn−1)∈Jsuch thatx˜=(#˜J1, . . . ,#˜^Jn−1). We can takeJn⊂ {v₁ⁿ, . . . , vⁿ_m

n}such thatV₁+. . .+V_n−1+ ^Jn =V₁+. . .+V_nandJ=(˜J1, . . . ,˜^Jn−1,Jn)∈J. Notice that x=J (n). This finishes the proof of the claim.

To finish the prove of the lemma, notice that for a pointed polyhedronP, we have P =conv.hull

x¹, . . . , x^r

+cone

y¹, . . . , y^t

wherexⁱ are the vertices ofP andyⁱ are its extremal rays (see[8, p. 107]); and we have necessarilyyⁱ 0 since P ⊂Rⁿ₊. 2

Remark 3.1.Notice thatJ (i) ∈P for all(J, i)∈J, hence we conclude fromLemma 2.4that P =conv.hullJ (i), (J, i)∈J

+cone{e1, . . . , e_n}. Conflict of interest statement

None declared.

Acknowledgements

We are grateful to P. Shmerkin for the useful discussions about the subject of this work, and to the anonymous referee for his excellent corrections and suggestions about this work.

References

[1]M. Hochman, P. Shmerkin, Local entropy averages and projections of fractal measures, Ann. Math. 175 (3) (2012) 1001–1059.

[2]R. Kaufman, On Hausdorff dimension of projections, Mathematika 15 (1968) 153–155.

[3]J.M. Marstrand, Some fundamental geometrical properties of plane sets of fractional dimensions, Proc. Lond. Math. Soc. (3) 4 (1954) 257–302.

[4]P. Mattila, Hausdorff dimension, orthogonal projections and intersections with planes, Ann. Acad. Sci. Fenn., Math. 1 (1975) 227–244.

[5]P. Mattila, Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability, Cambridge University Press, 1995.

[6]C.G. Moreira, J.-C. Yoccoz, Stable intersection of regular cantor sets with large Hausdorff dimensions, Ann. Math. 154 (1) (2001) 45–96.

[7]Y. Peres, W. Schalg, Smoothness of projections, Bernoulli convolutions, and the dimensions of exceptions, Duke Math. J. 102 (2) (2000) 193–251.

[8]A. Schrijver, Theory of Linear and Integer Programming, Wiley–Interscience, Chichester, 1986.