Furstenberg sets for a fractal set of directions

arXiv:1009.0481v1 [math.CA] 2 Sep 2010 FURSTENBERG SETS FOR A FRACTAL SET OF DIRECTIONS URSULA MOLTER AND EZEQUIEL RELA Abstract. In this note we study the behavior of the size of Furstenberg sets with respect to the size of the set of directions defining it. For any pair α, β ∈(0,1], we will say that a set E⊂R2is an Fαβ -set if there is a subset Lof the unit circle of Hausdorff dimension at least β and, for each direction ein L, there is a line segment ℓein the direction of esuch that the Hausdorff dimension of the set E∩ℓeis equal or greater than α. The problem is considered in the wider scenario of generalized Hausdorff measures, giving estimates on the appropriate dimension functions for each class of Furstenberg sets. As a corollary of our main results, we obtain that dim(E)≥max α+β 2; 2α+β−1 for any E∈Fαβ. In particular we are able to extend previously known results to the “endpoint” α= 0 case. 1. Introduction In this article we are interested in the study of dimension properties of Furstenberg sets associated to fractal sets of directions. Let us introduce the definition of our object of study. In the sequel, we will denote with dim(E) the Hausdorff dimension of the set E. Definition 1.1. For α, β in (0,1], a subset Eof R2will be called an Fαβ-set if there is a subset Lof the unit circle such that dim(L)≥βand, for each direction ein L, there is a line segment ℓein the direction of esuch that the Hausdorff dimension of the set E∩ℓeis equal or greater than α. This generalizes the classical definition of Furstenberg sets, when the whole circle is considered as set of directions. For L=S, which is a particular case of β= 1, we recover the classical class Fαof α-Furstenberg sets, and the best known result is (1) max α+1 2; 2α≤γ(α)≤1 2+3 2α, α ∈(0,1]. where γ(α) = inf{dim(E) : E∈Fα}. In [MR10] and [MR] the above inequalities are proved in the general setting of dimension functions, allowing the extension to the endpoint α= 0 for some class of generalized Furstenberg sets. Unavoidable references on this matter are [Wol99], [Wol03], [KT01] and [Tao]. 1991 Mathematics Subject Classification. Primary 28A78, 28A80. Key words and phrases. Furstenberg sets, Hausdorff dimension, dimension function, Kakeya sets. This research is partially supported by Grants: PICT2006-00177, PIP 11220080100398 and UBACyT X149. 1 2 URSULA MOLTER AND EZEQUIEL RELA The purpose of this note is to study how the parameter βaffects the bounds above. Moreover, by using general Hausdorff measures, we will extend the inequalities (1) to the zero dimensional case. From our results we will derive the following proposition. Proposition 1.2. For any set E∈Fαβ, we have that (2) dim(E)≥max α+β 2; 2α+β−1, α, β > 0. It is not hard to prove Proposition 1.2 directly, but we will study this problem in a wider scenario and derive it as a corollary. We also remark that our results are consistent with the ones in [Mit02], where the author proves, essentially, the second bound for the case α= 1, β∈(0,1]. There is a natural way to generalize this problem by looking at dimension functions that are not necessarily power functions ([Hau18]). Let us begin with the notion of dimension functions. 1.1. Dimension Functions. Definition 1.3. The following class of functions will be called dimension functions. H:= {h: [0,∞)→[0 : ∞),non-decreasing, right continuous, h(0) = 0}. The important subclass of those h∈Hthat satisfy a doubling condition will be denoted by Hd: Hd:= {h∈H:h(2x)≤Ch(x) for some C > 0}. Remark 1.4. Clearly, if h∈Hd, the same inequality will hold (with some other constant) if 2 is replaced by any other λ > 1. We also remark that any concave function trivially belongs to Hd. Also note that the monotonicity of himplies that C≥1. If one only looks at the power functions, there is a natural total order given by the exponents. If we denote with hα(x) = xα, then hαis, in some sense, smaller than hβif and only if α < β. In Hwe also have a natural notion of order, but we can only obtain a partial order. Definition 1.5. Let g, h be two dimension functions. We will say that gis dimensionally smaller than hand write g≺hif and only if lim x→0+ h(x) g(x)= 0. We also remark that we will be particularly interested in the special subclass of dimension functions that allows us to classify zero dimensional sets, that means, that his in this class if it is smaller than any of the functions xα,α > 0. Definition 1.6. A function h∈Hwill be called “zero dimensional dimension function” if h≺hαfor any α > 0. We will denote by H0the subclass of those functions. FURSTENBERG SETS FOR A FRACTAL SET OF DIRECTIONS 3 As usual, the h-dimensional (outer) Hausdorff measure Hhwill be defined as follows. For a set E⊆Rnand δ > 0, write Hh δ(E) = inf (X i h(diam(Ei)) : E⊂∞ [ i Ei,diam(Ei)< δ). The h-dimensional Hausdorff measure Hhof Eis defined by Hh(E) = sup δ>0Hh δ(E). We remark that, even though they would not lead to the exact same measures, we will consider functions g, h such that there exist constants c, C with 0 < c ≤g(x) h(x)≤C < ∞for all x > 0 to be equivalent. In that case we write g≡h. To measure the “distance” between to dimension functions, we introduce the following notion: Definition 1.7. Let g, h ∈Hwith g≺h. Define the “gap” between gand has (3) ∆(x) = h(x) g(x). From this definition and the definition of partial order, we always have that limx→0∆(x) = 0, and therefore the speed of convergence to zero can be seen as a notion of distance between gand h. Now we present the problem. Let us begin with the definition of Fhg-sets. Let hand gbe two dimension functions. A set E⊆R2is a Furstenberg set of type hg, or an Fhg-set, if there is a subset Lof the unit circle such that Hg(L)>0 and, for each direction ein L, there is a line segment ℓein the direction of esuch that Hh(ℓe∩E)>0. Note that this hypothesis is stronger than the one used to define the original Furstenberg-αsets. However, the hypothesis dim(E∩ℓe)≥αis equivalent to Hβ(E∩ℓe)>0 for any βsmaller than α. If we use the wider class of dimension functions introduced above, the natural way to define Fhsets would be to replace the parameters β < α with two dimension functions satisfying the relation h≺h. But requiring E∩ℓeto have positive Hh measure for any h≺himplies that it has also positive Hhmeasure (Theorem 42, [Rog70]). Therefore, this definition is the natural generalization of the F+ αβ class defined below. Definition 1.8. For each pair α, β in (0,1], a subset Eof R2will be called an F+ αβ-set if there is a subset Lof the unit circle such that Hβ(L)>0 and, for each direction ein L, there is a line segment ℓein the direction of esuch that Hα(ℓe∩E)>0. Now, for the sake of clarity in the proof of our results, we will perform the same reduction made in [MR10]. A standard pigeonhole argument allows us to work with the following definition. Definition 1.9. Let hand gbe two dimension functions. A set E⊆R2 is a Furstenberg set of type hg, or an Fhg-set, if there is a subset Lof the unit circle such that Hg(L)>0 and, for each direction ein L, there is a line 4 URSULA MOLTER AND EZEQUIEL RELA segment ℓein the direction of esuch that Hh δ(ℓe∩E)>1 for all δ < δEfor some δE>0 with δEdepending only on E. Following the intuition suggested by Proposition 1.2, one could conjecture that if Ebelong to the class Fhg then an appropriate dimension function for Eshould be dimensionally greater than h2g id and h√g(where id is the identity function). This will indeed be the case, and we will provide some estimates on the gap between those conjectured dimension functions and a generic test function h∈Hto ensure that Hh(E)>0. In addition we illustrate with some examples. We will consider the two results separately. Namely, for a given pair of dimension functions g∈Hand h∈Hd, in Section 3 we obtain sufficient conditions on a test dimension function h∈H,h≻h2g id to ensure that Hh(E)>0 for any set E∈Fhg. In Section 4 we consider the analogous problem for h≻h√g. The next section summarizes some preliminary results to be used in our proofs and additional notation. Finally, in Section 5 we briefly discuss the appropriate notion of size for the set of directions defining the Furstenberg classes. 2. Preliminaries In this section we include some preliminary and technical results needed in the sequel. We will use the notation A.Bto indicate that there is a constant C > 0 such that A≤CB, where the constant is independent of A and B. By A∼Bwe mean that both A.Band B.Ahold. As usual, by aδ-covering of a set Ewe mean a covering of Eby sets Uiwith diameters not exceeding δ. In Section 3 the main tool will be an L2estimate for the Kakeya maximal function for general measures. For an integrable function on Rn, the Kakeya maximal function at scale δwill be Kδ(f) : Sn−1→R, Kδ(f)(e) = sup x∈Rn 1 |Tδ e(x)|ZTδ e(x)|f(x)|dx e ∈Sn−1, where Tδ e(x) is a 1 ×δ-tube (by this we mean a tube of length 1 and cross section of radius δ) centered at xin the direction e. The estimate we need is the main result of [Mit02]. There the author proves (Theorem 3.1) the following. Proposition 2.1. Let µbe a Borel probability measure on Ssuch that µ(B(x, r)) .ϕ(r)for some non-negative function ϕfor all r≪1. Define the Kakeya maximal operator Kδas usual: Kδ(f)(e) = sup x∈Rn 1 |Tδ e(x)|ZTδ e(x)|f(x)|dx, e ∈Sn−1. Then we have the estimate (4) kKδk2 L2(R2)→L2(S,dµ).C(δ) = Z1 δ ϕ(u) u2du. Remark 2.2. It should be noted that if we choose ϕ(x) = xs, then we obtain as a corollary that (5) kKδk2 L2(R2)→L2(S,dµ).δs−1. FURSTENBERG SETS FOR A FRACTAL SET OF DIRECTIONS 5 In the special case of s= 1, the bound has the known logarithmic growth: kKδk2 L2(R2)→L2(S,dµ)∼log(1 δ). This result will be used in Section 3, where the hypotheses imposed on a set Efor being an Fhg set guarantee, via Frostman’s lemma, that there exists a probability measure µon the set of directions Lwith µ(Br).g(r) for any ball Br(see [Mat95]). Let us remark that (5) suggests that the constant C(δ) plays, in the general case, the role of g id (δ). In Section 4 we perform a more combinatorial kind of proof. We introduce the notion of δ-entropy of a set Ein the next definition Definition 2.3. Let E⊂Rnand δ∈R>0. The δ-entropy of Eis the maximal possible cardinality of a δ-separated subset of E. We will denote this quantity with Nδ(E). The main idea is to relate the δ-entropy to some notion of size of the set. Clearly, the entropy is essentially the Box dimension or the Packing dimension of a set (see [Mat95] or [Fal03] for the definitions) since both concepts are defined in terms of separated δballs with centers in the set. However, for our proof we will need to relate the entropy of a set to some quantity that has the property of being (in some sense) stable under countable unions. One choice is therefore the notion of Hausdorff content, which enjoys the needed properties: it is an outer measure, is finite, and reflects the entropy of a set in the following manner. Recall that the g-dimensional Hausdorff content of a set Eis defined as (6) Hg ∞(E) = inf (X i g(diam(Ui) : E⊂[ i Ui). Note that the g-dimensional Hausdorff content Hg ∞is clearly not the same than the g-dimensional Hausdorff measure Hg. In fact, they are the measures obtained by applying Method I and Method II (see [Mat95]) respectively to the premeasure that assigns to a set Athe value g(diam(A)). For future reference, we state the following estimate for the δ-entropy of a set with positive g-dimensional Hausdorff content as a lemma. Lemma 2.4. Let g∈Hand let Abe any set. Let Nδ(A)be the δ-entropy of A. Then Nδ(A)≥Hg ∞(A) g(δ). Proof. Let {xi}N i=1 be a maximal δ-separated subset. By maximality, we can cover Awith balls B(xi, δ). Therefore, for the g-dimensional Hausdorff content Hg ∞, we have the bound (7) Hg ∞(A)≤ N X iHg ∞(B(xi, δ)) ≤Ng(δ) and it follows that Nδ(A)≥N≥Hg ∞(A) g(δ). Of course, this result is meaningful when Hg ∞(A)>0. We will use it in the case Hg(A)>0 which is equivalent to Hg ∞(A)>0. For a detailed study of the properties of Hgand Hg ∞see [Del02] and [Del03]. 6 URSULA MOLTER AND EZEQUIEL RELA Note that the lemma above only requires the finiteness and the subadditivity of the Hausdorff content. The relevant feature that will be needed in our proof is the σ-subadditivity, which is a property that the Box dimension does not share. Now we introduce the following notation and a technical lemma. Definition 2.5. Let b={bk}k∈Nbe a decreasing sequence with lim bk= 0. For any family of balls B={Bj}with Bj=B(xj;rj), rj≤1, and for any set E, we define (8) Jb k:= {j∈N:bk< rj≤bk−1}, and (9) Ek:= E∩[ j∈Jb k Bj. In the particular case of the dyadic scale b={2−k}, we will omit the superscript and denote (10) Jk:= {j∈N: 2−k< rj≤2−k+1}. The next lemma introduces a technique used in [MR10] to decompose the set of all directions. Lemma 2.6. Let Ebe an Fhg-set for some h,g∈Hwith the directions in L⊂Sand let a={ak}k∈N∈ℓ1be a non-negative sequence. Let B={Bj} be a δ-covering of Ewith δ < δEand let Ekand Jkbe as above. Define Lk:= e∈S:Hh δ(ℓe∩Ek)≥ak 2kak1. Then L=∪kLk. The proof follows directly from the summability of a. 3. The Kakeya type bound In this section we prove a generalized version of the announced bound dim(E)≥2α+β−1 for E∈Fαβ. We have the following theorem. Theorem 3.1 (hg →h2g id ).Let g∈H,h∈Hdbe two dimension functions and let Ebe an Fhg-set. For δ > 0, let C(δ)be as in (4). For any h∈H such that X kqh2(2−k)C(2−k) h(2−k)<∞,Hh(E)>0. Proof. Let E∈Fhg and let {Bj}j∈Nbe a covering of Eby balls with Bj=B(xj;rj). We need to bound Pjh(2rj) from below. Since his nondecreasing, it suffices to obtain the bound (11) X j h(rj)&1 for any h∈Hsatisfying the hypothesis of the theorem. Define a={ak}by a2 k=h2(2−k)C(2−k) h(2−k). Also define, as in the previous section, for each k∈N,Jk={j∈N: 2−k< rj≤2−k+1}and Ek= FURSTENBERG SETS FOR A FRACTAL SET OF DIRECTIONS 7 E∩∪j∈JkBj. Since by hypothesis a∈ℓ1, we can apply Lemma 2.6 to obtain the decomposition of the set of directions as L=SkLkassociated to this choice of a. We will apply the maximal function inequality to a weighted union of indicator functions. For each k, let Fk=[ j∈Jk Bjand define the function f:= h(2−k)2kχFk. We will use the L2norm estimates for the maximal function. We can compute directly the L2norm of f: kfk2 2=h2(2−k)22kZ∪JkBj dx .h2(2−k)22kX j∈Jk r2 j .h2(2−k)#Jk, since rj≤2−k+1 for j∈Jk. Therefore (12) kfk2 2.#Jkh2(2−k). The same arguments used in the proof of Theorem 3.1 in [MR10] allows us to obtain a lower bound for the maximal function. Essentially, the maximal function is pointwise bounded from below by the average of fover the tube centered on the line segment ℓefor any e∈Lk. Therefore, we have the following bound for the (L2, µ) norm. Here, µis a measure supported on L that obeys the law µ(B(x, r)≤g(r) for any ball B(x, r) given by Frostman’s lemma. (13) kK2−k+1 (f)k2 L2(dµ)&a2 kµ(Lk) = µ(Lk)h2(2−k)C(2−k) h(2−k). Combining (13) with the maximal inequality (4), we obtain µ(Lk)h2(2−k)C(2−k) h(2−k).kK2−k+1 (f)k2 2.C(2−k+1)kfk2 2≤C(2−k)kfk2 2. We also have the bound (12), which implies that µ(Lk) h(2−k).#Jk. Now we are able to estimate the sum in (11). Let hbe a dimension function satisfying the hypothesis of Theorem 3.1. We have X j h(rj)≥X k h(2−k)#Jk &X k µ(Lk)≥µ(L)>0.  Corollary 3.2. Let Ean F+ αβ-set. If his any dimension function satisfying (14) h(x)≥Cx2α+β−1logθ(1 x) 8 URSULA MOLTER AND EZEQUIEL RELA for θ > 2, then Hh(E)>0. Proof. It follows directly, since in this case we have C(δ).δβ−1, and therefore the sum in Theorem 3.1 is X ksh2(2−k)C(2−k) h(2−k).X ks2−k2α2−k(β−1) h(2−k) ≤X ks2−k(2α+β−1) (2−k)2α+β−1logθ(2k) =X k 1 kθ 2 <∞.  Remark 3.3. Note that the bound dim(E)≥2α+β−1 for E∈Fαβ follows directly from this last corollary. 4. The combinatorial bound In this section we deal with the bound hg →h√g, which is the significant bound near the endpoint α=β= 0 and generalizes the bound dim(E)≥ β 2+αfor E∈Fαβ. Note that the second bound in (2) is meaningless for small values of αand β. We will consider separately the cases of h being zero dimensional or positive dimensional. In the next theorem, the additional condition on hreflects the positivity of the dimension function. We believe that it would be helpful to cite, without the proofs, two relevant lemmas used in [MR10]. The first is a “splitting lemma”, which says that a linear set with positive h-dimensional mass can be splitted into two well separated linear subsets. Lemma 4.1. Let h∈H,δ > 0,Ian interval and E⊆I. Let η > 0be such that h−1(η 8)< δ and Hh δ(E)≥η > 0. Then there exist two subintervals I−, I+that are h−1(η 8)-separated and with Hh δ(I±∩E)&η. The second lemma is the combinatorial ingredient in the proof of both Theorem 4.3 and Theorem 4.6. This lemma provides an estimate on the number of lines with certain separation that intersect two balls of a given size. Lemma 4.2. Let b={bk}k∈Nbe a decreasing sequence with lim bk= 0. Given a family of balls B={B(xj;rj)}, we define Jb kas in (8) and let {ei}Mk i=1 be a bk-separated set of directions. Assume that for each ithere are two line segments I+ eiand I− eilying on a line in the direction eithat are sk-separated for some given skDefine Πk=Jb k×Jb k×{1, .., Mk}and Lb kby Lb k:= (j+, j−, i)∈Πk:I− ei∩Bj−6=∅I+ ei∩Bj+6=∅. If 1 5sk> bk−1for all k, then #Lb k.bk−1 bk 1 sk#Jb k2. FURSTENBERG SETS FOR A FRACTAL SET OF DIRECTIONS 9 With these two lemmas we are now ready to prove the main result of this section. We have the following theorem. Recall that hα(x) = xα. Theorem 4.3 (hg →h√g,h≻hα).Let g∈H,h∈Hdbe two dimension functions such that h(x).xαfor some 0< α < 1and let Ebe an Fhg-set. Let h∈Hwith h≺h√g. If X kh(2−k)√g(2−k) h(2−k)2α 2α+1 <∞, then Hh(E)>0. Proof. Let E∈Fhg and let {Bj}j∈Nbe a covering of Eby balls with Bj= B(xj;rj). Define ∆ = h√g hand consider the sequence a=n∆2α 2α+1 (2−k)ok. Also define, as in the previous section, for each k∈N,Jk={j∈N: 2−k< rj≤2−k+1}and Ek=E∩ ∪j∈JkBj. Since by hypothesis a∈ℓ1, we can apply Lemma 2.6 to obtain the decomposition of the set of directions as L=SkLkassociated to this choice of a, where Lkis defined as Lk:= e∈S:Hh δ(ℓe∩Ek)≥ak 2kak1. We can apply Lemma 4.1 with η=ak 2kak1to ℓe∩Ek. Therefore we obtain two intervals I− eand I+ e, contained in ℓewith Hh δ(I± e∩Ek)&ak that are h−1(rak)-separated for r=1 16kak1. Now, let {ek j}Nk j=1 be a 2−k-separated subset of Lk. Taking into account the estimate for the entropy given in Lemma 2.4. We obtain then that (15) Nk&Hg ∞(Lk) g(2−k). Define Πk:= Jk×Jk×{1, .., Nk}and (16) Tk:= (j−, j+, i)∈Πk:I− ei∩Ek∩Bj−6=∅I+ ei∩Ek∩Bj+6=∅. The idea is to count the elements of Tkin two ways. If we fix a pair j−and j+and count for how many values of ithe triplet (j−, j+, i) belongs to Tk, we obtain, by using Lemma 4.2 for the choice b={2−k}, that (17) #Tk.1 h−1(rak)(#Jk)2. Second, fix i. In this case, we have by hypothesis that Hh δ(I+ ei∩Ek)&ak, so Pj+h(rj+)&ak. Therefore, ak.X (j−,j+,i)∈Tk h(rj+)≤Kh(2−k), where Kis the number of elements of the sum. Therefore K&ak h(2−k). The same holds for j−, so (18) #Tk&Nkak h(2−k)2 .