Improving dimension estimates for Furstenberg-type sets

arXiv:0904.2710v2 [math.CA] 18 Nov 2009 IMPROVING DIMENSION ESTIMATES FOR FURSTENBERG-TYPE SETS URSULA MOLTER AND EZEQUIEL RELA Abstract. In this paper we study the problem of estimating the generalized Hausdorff dimension of Furstenberg sets in the plane. For α∈(0,1], a set Fin the plane is said to be an α-Furstenberg set if for each direction ethere is a line segment ℓein the direction of efor which dimH(ℓe∩F)≥α. It is well known that dimH(F)≥max{2α, α +1 2}- and it is also known that these sets can have zero measure at their critical dimension. By looking at general Hausdorff measures Hhdefined for doubling functions, that need not be power laws, we obtain finer estimates for the size of the more general h-Furstenberg sets. Further, this approach allow us to sharpen the known bounds on the dimension of classical Furstenberg sets. The main difficulty we had to overcome, was that if Hh(F) = 0, there always exists g≺hsuch that Hg(F) = 0 (here ≺refers to the natural ordering on general Hausdorff dimension functions). Hence, in order to estimate the measure of general Furstenberg sets, we have to consider dimension functions that are a true step down from the critical one. We provide rather precise estimates on the size of this step and by doing so, we can include a family of zero dimensional Furstenberg sets associated to dimension functions that grow faster than any power function at zero. With some additional growth conditions on these zero dimensional functions, we extend the known inequalities to include the endpoint α= 0. 1. Introduction In this paper we study dimension properties of sets of Furstenberg type. We are able to sharpen the known bounds about the Hausdorff dimension of these sets using general doubling dimension functions for the estimates. Let us recall the notion of Furstenberg sets. For αin (0,1], a subset E of R2is called Furstenberg set or Fα-set if for each direction ein the unit circle there is a line segment ℓein the direction of esuch that the Hausdorff dimension of the set E∩ℓeis equal or greater than α. We will also say that such set Ebelongs to the class Fα. It is known ([18], see also [17], [19], [20], [7], [16] for related topics and [8], [15] for a discretized version of this problem) that for any Fα-set E⊆R2the Hausdorff dimension (dim(E)) must satisfy the inequality dim(E)≥max{2α, α +1 2}and there are examples of Fα-sets Ewith dim(E)≤1 2+3 2α. If we denote by γ(α) = inf{dim(E) : E∈Fα}, 1991 Mathematics Subject Classification. Primary 28A78, 28A80. Key words and phrases. Furstenberg sets, Hausdorff dimension, dimension function. This research is partially supported by Grants: PICT2006-00177 and UBACyT X149. 1 2 URSULA MOLTER AND EZEQUIEL RELA then (1) max{α+1 2; 2α} ≤ γ(α)≤1 2+3 2α, α ∈(0,1]. In this paper we study a more general notion of Furstenberg sets. To that end we will use a finer notion of dimension already defined by Hausdorff [6]. Definition 1.1. 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 1.2. 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. As usual, the h-dimensional (outer) Hausdorff measure Hhwill be defined as follows. For a set E⊆R2and δ > 0, write Hh δ(E) = inf (X i h(diam(Ei)) : E⊂∞ [ i Ei,diam(Ei)< δ). Then the h-dimensional Hausdorff measure Hhof Eis defined by Hh(E) = sup δ>0Hh δ(E). This notion generalizes the classical α-Hausdorff measure to functions hthat are different to xα. It is well known that a set of Hausdorff dimension α can have zero, positive or infinite α-dimensional measure. The desirable situation, in general, is to work with a set which is truly α-dimensional, that is, it has positive and finite α-dimensional measure. In this case we refer to this set as an α-set. Now, given an α-dimensional set Ewithout this last property, one could expect to find in the class Han appropriate function hto detect the precise “size” of it. By that we mean that 0 <Hh(E)<∞, and in this case Eis referred to as an h-set. We mention one example: A Kakeya set is a compact set containing a unit segment in every possible direction. It is known that there are Kakeya sets of zero measure and it is conjectured that they must have full Hausdorff dimension. The conjecture was proven by Davies [2] in R2and remains open for higher dimensions. Since in the class of planar Kakeya sets there are several distinct types of two dimensional sets (i.e. with positive or null Lebesgue measure), one would like to associate a dimension function to the whole class. A dimension function h∈Hwill be called the exact Hausdorff dimension function of the class of sets Cif •For every set Ein the class C,Hh(E)>0. •There are sets E∈ C with Hh(E)<∞. IMPROVING DIMENSION ESTIMATES FOR FURSTENBERG-TYPE SETS 3 In the direction of finding the exact dimension of the class of Kakeya sets in R2, Keich [9] has proven that in the case of the Minkowsky dimension the exact dimension function is h(x) = x2log(1 x). For the case of the Hausdorff dimension, he provided some partial results. Specifically, he shows that in this case the exact dimension function hmust decrease to zero at the origin faster than x2log(1 x) log log(1 x)2+εfor any given ε > 0, but slower than x2log(1 x). This notion of speed of convergence to zero will allow us to define a partial order between dimension functions that extends the usual order on the power laws (see 1.3). In this paper we are interested in the problem of studying the exact Hausdorff dimension of the class of Furstenberg-type sets (for the precise definition, see 1.5). We are able to find lower bounds for the dimension function, i.e. for a given class of Furstenberg-type sets, we find a dimension function hwith the property that any set in the class has positive Hhmeasure. For the construction of h-sets associated to certain sequences see the work of Cabrelli et al [1] (see also [5]). We refer to the work of Olsen and Renfro [13], [12], [11] for a detailed study of the exact Hausdorff dimension of the Liouville numbers L, which is a known example of a zero dimensional set. Moreover, the authors prove that this is also a dimensionless set, i.e. there is no h∈Hsuch that 0 <Hh(L)<∞(equivalently, for any dimension function h, one has Hh(L)∈ {0,∞}). In that direction, further improvements are due to Elekes and Keleti [3]. There the authors prove much more than that there is no exact Hausdorff-dimension function for the set Lof Liouville numbers: they prove that for any translation invariant Borel measure Lis either of measure zero or has non-sigma-finite measure. So in particular they answer the more interesting question that there is no exact Hausdorff-dimension function for Leven in the stronger sense when requiring only sigma-finiteness instead of finiteness. If one only looks at the power functions, there is a natural total order given by the exponents. In Hwe also have a natural notion of order, but we can only obtain a partial order. Definition 1.3. 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. Remark 1.4. Note that this partial order, restricted to the class of power functions, recovers the natural order mentioned above. That is, xα≺xβ⇐⇒ α < β. Now we can make a precise statement of the problem. We begin with the definition of the Furstenberg-type sets. Definition 1.5. Let hbe a dimension function. A set E⊆R2is a Furstenberg set of type h, or an Fh-set, if for each direction e∈Sthere 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 4 URSULA MOLTER AND EZEQUIEL RELA 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 Fh-sets would be to replace the parameters β < α with two dimension functions satisfying the relation h≺h. But requiring E∩ℓeto have positive Hhmeasure for any h≺himplies that it has also positive Hhmeasure (Theorem 42, [14]). Due to the existence of Fα-sets with Hα(E∩ℓe) = 0 for each e, it will be useful to introduce the following subclass of Fα: Definition 1.6. A set E⊆R2is an F+ α-set if for each e∈Sthere is a line segment ℓesuch that Hα(ℓe∩E)>0. Remark 1.7. Given an Fh-set Efor some h∈H, it is always possible to find two constants mE, δE>0 and a set ΩE⊆Sof positive σ-measure such that Hh δ(ℓe∩E)> mE>0∀δ < δE,∀e∈ΩE. For each e∈S, there is a positive constant mesuch that Hh(ℓe∩E)> me. Now consider the following pigeonholing argument. Let Λn={e∈S: 1 n+1 ≤me<1 n}. At least one of the sets must have positive measure, since S=∪nΛn. Let Λn0be such set and take 0 <2mE<1 n0+1 . Hence Hh(ℓe∩E)>2mE>0 for all e∈Λn0Finally, again by pigeonholing, we can find ΩE⊆Λn0of positive measure and δE>0 such (2) Hh δ(ℓe∩E)> mE>0∀e∈ΩE∀δ < δE. To simplify notation throughout the paper, since inequality (2) holds for any Furstenberg set and we will only use the fact that mE,δEand σ(ΩE) are positive, it will be enough to consider the following definition of Fh-sets: Definition 1.8. Let hbe a dimension function. A set E⊆R2is Furstenberg set of type h, or an Fh-set, if for each e∈Sthere is a line segment ℓein the direction of esuch that Hh δ(ℓe∩E)>1 for all δ < δEfor some δE>0. The purpose of this paper is to obtain an estimate of the dimension of an Fh-set. By analogy to the classical estimate (1), we first note that if his a general dimension function (not xα), α+1 2translates to h√·and 2αto h2. Hence, when aiming to obtain an estimate of the Hausdorff measure of our set E, the naive approach would be to prove that if a dimension function h satisfies (3) h≺h2or h≺h√·, then Hh(E)>0. However, there is no hope to obtain such a general result, since for the special case of the identity function h(x) = x, this requirement would contradict (again by Theorem 42, [14]) the existence of zero measure planar Kakeya sets. Therefore, it is clear that one needs to take a step down from the conjectured dimension function. The main result of this paper is to show that this step does not need to be as big as a power. It can be, for example, just the power of a log. Precisely, we find conditions on the step that guarantee IMPROVING DIMENSION ESTIMATES FOR FURSTENBERG-TYPE SETS 5 lower bounds on the dimension of Fh-sets. Further, our techniques allow us to analyze Furstenberg-type sets of Hausdorff dimension zero. This can be done considering dimension functions hthat are smaller than xαfor any α > 0. To keep the analogy with the classical Furstenberg sets, we will introduce the following notation: Definition 1.9. Given two dimension functions g, h ∈H, we define the following quotients which are related to the step-size between two functions: ∆0(g, h)(x) := ∆0(x) = g(x) h(x)∆1(g, h)(x) := ∆1(x) = g(x) h2(x). When proving the first case of the inequalities in (3), the relevant quotient is ∆1, which gives the (better) bound dim ≥2αin the classical case at the endpoint α= 1. At the other endpoint, α= 0, the best bound is dim ≥α+1 2 and the quotient to analyze here in our generalized problem is ∆0. This paper is organized as follows. In Section 2, we introduce some further notation and prove a preliminary lemma to be used in the remainder of the paper. In Section 3 we prove the h2bound, in Section 4 the h√·bound under some positivity assumptions on the function hand in Section 5 we drop this last condition to obtain a partial result on the zero dimensional Furstenberg sets. In addition we discuss our methods and study the Furstenberg problem in the extreme case of the counting measure. This is, roughly speaking, the case h≡1. 2. Remarks, notation and more definitions 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 Aand B. By A∼Bwe mean that both A.Band B.Ahold. On the circle Swe consider the arclength measure σ. By L2(S) we mean L2(S, dσ). For each e∈S,ℓewill be a unit line segment in the direction e. As usual, by a δ-covering of a set Ewe mean a covering of Eby sets Uiwith diameters not exceeding δ. We introduce the following notation: Definition 2.1. 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 (4) Jb k:= {j∈N:bk< rj≤bk−1}, and Ek:= E∩[ j∈Jb k Bj. In the particular case of the dyadic scale b={2−k}, we will omit the superscript and denote (5) Jk:= {j∈N: 2−k< rj≤2−k+1} The next lemma introduces a technique we borrow from [18] to decompose the set of all directions. 6 URSULA MOLTER AND EZEQUIEL RELA Lemma 2.2. Let Ebe an Fh-set for some h∈Hand a={ak}k∈N∈ℓ1a non-negative sequence. Let B={Bj}be a δ-covering of Ewith δ < δEand let Ekand Jkbe as above. Define Ωk:= e∈S:Hh δ(ℓe∩Ek)≥ak 2kak1. Then S=∪kΩk. Proof. Clearly Ωk⊂S. To see why S=∪kΩk, assume that there is a direction e∈Sthat is not in any of the Ωk. Then for that direction we would have that 1<Hh δ(ℓe∩E)≤X kHh δ(ℓe∩E∩[ j∈Jk Bj)≤X k 1ak 2kak1 =1 2, which is a contradiction.  As a final remark we note that in the following sections our aim will be to prove essentially (6) X j h(rj)&1, provided that his a small enough dimension function. The idea will be to use the dyadic partition of the covering to obtain that X j h(rj)&∞ X k=0 h(2−k)#Jk. The lower bounds we need will be obtained if we can prove lower bounds on the quantity Jkin terms of the function hbut independent of the covering. 3. The h→h2bound In this section we generalize the first inequality of (1), that is, dim(E)≥ 2αfor any Fα-set. For this, given a dimension function h≺h2, we impose some sufficient growth conditions on the gap ∆1(x) := h(x) h2(x)to ensure that Hh(E)>0. We have the following theorem: Theorem 3.1. Let h∈Hdbe a dimension function and let Ebe an Fh-set. Let h∈Hsuch that h≺h2. If X k h(2−k)qk h(2−k)<∞, then Hh(E)>0. The main tool for the proof of this theorem will be an L2bound for the Kakeya maximal function on R2. For an integrable function on Rn, the Kakeya maximal function at scale δwill be f∗ δ:Sn−1→R, f∗ δ(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. It is well known that in R2the Kakeya maximal function satisfies the bound (see [18]) IMPROVING DIMENSION ESTIMATES FOR FURSTENBERG-TYPE SETS 7 (7)  f∗ δ 2 2.log(1 δ)kfk2 2. It is also known that the log growth is necessary (see [9]), because of the existence of Kakeya sets of zero measure in R2. See also [10] for estimates on the Kakeya maximal function with more general measures on the circle. We now prove Theorem 3.1. We remark that since this theorem says, roughly speaking, that the dimension of an Fh-set should be about h2, the step down must be taken from this dimension function. This is the role played by ∆1(h, h2)(x) = h(x) h2(x)in this section. Proof. By 1.8, since E∈Fh, we have Hh δ(ℓe∩E)>1 for all e∈Sand for any δ < δE. Let {Bj}j∈Nbe a covering of Eby balls with Bj=B(xj;rj). We need to bound Pjh(2rj) from below. Since his non-decreasing, it suffices to obtain the bound (8) X j h(rj)&1 for any h∈Hsatisfying the hypothesis of the theorem. Clearly we can restrict ourselves to δ-coverings with δ < δE 5. Define a={ak}with ak=qk ∆1(2−k). 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 2.2 to obtain the decomposition S=SkΩkassociated 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. The L2 norm of fcan be easily estimated as follows: 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. Hence, (9) kfk2 2.#Jkh2(2−k). Now fix kand consider the Kakeya maximal function f∗ δof level δ= 2−k+1 associated to the function fdefined for this value of k. 8 URSULA MOLTER AND EZEQUIEL RELA In Ωkwe have the following pointwise lower estimate for the maximal function. Let ℓebe the line segment such that Hh δ(ℓe∩E)>1, and let Tebe the rectangle of width 2−k+2 around this segment. Define, for each e∈Ωk, (10) Jk(e) := {j∈Jk:ℓe∩E∩Bj6=∅}. With the aid of the Vitali covering lemma, we can select a subset of disjoint balls e Jk(e)⊆Jk(e) such that [ j∈Jk(e) Bj⊆[ j∈ e Jk(e) B(xj; 5rj). Note that every ball Bj,j∈Jk(e), intersects ℓeand therefore at least half of Bjis contained in the rectangle Te, yielding |Te∩Bj| ≥ 1 2πr2 j. Hence, by definition of the maximal function, using that rj≥2−k+1 for j∈Jk(e), |f∗ 2−k+1 (e)| ≥ 1 |Te|ZTe f dx =h(2−k)2k |Te|Te∩∪Jk(e)Bj &h(2−k)22kTe∩∪e Jk(e)Bj &h(2−k)22kX j∈ e Jk(e) r2 j &h(2−k)# e Jk(e) &X e Jk(e) h(rj). Now, since (11) ℓe∩Ek⊆[ j∈Jk(e) Bj⊆[ j∈ e Jk(e) B(xj; 5rj) and for e∈Ωkwe have Hh δ(ℓe∩Ek)&ak, we obtain |f∗ 2−k+1 (e)|&X e Jk(e) h(rj)&X j∈ e Jk(e) h(5rj)&ak. Therefore we have the estimate (12) kf∗ 2−k+1 k2 2&ZΩk|f∗ 2−k+1 (e)|2dσ &a2 kσ(Ωk) = σ(Ωk)k ∆1(2−k). Combining (9), (12) and using the maximal inequality (7), we obtain σ(Ωk)k ∆1(2−k).kf∗ 2−k+1 k2 2.log(2k)kfk2 2.k#Jkh2(2−k), and therefore σ(Ωk) h(2−k).#Jk. IMPROVING DIMENSION ESTIMATES FOR FURSTENBERG-TYPE SETS 9 Now we are able to estimate the sum in (8). 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 σ(Ωk)≥σ(S)>0.  Applying this theorem to the class F+ α, we obtain a sharper lower bound on the generalized Hausdorff dimension: Corollary 3.2. Let Ean F+ α-set. If his any dimension function satisfying h(x)≥Cx2αlog1+θ(1 x)for θ > 2then Hh(E)>0. Remark 3.3. At the endpoint α= 1, this estimate is worse than the one due to Keich. He obtained, using strongly the full dimension of a ball in R2, that if Eis an F+ 1-set and his a dimension function satisfying the bound h(x)≥Cx2log(1 x)log log(1 x)θfor θ > 2, then Hh(E)>0. Remark 3.4. Note that the proof above relies essentially on the L1and L2size of the ball in R2, not on the dimension function h. Moreover, we only use the “gap” between hand h2(measured by the function ∆1). This last observation leads to conjecture that this proof can not be used to prove that an Fh-set has positive h2measure, since in the case of h(x) = x, as we remarked in the introduction, this would contradict the existence of Kakeya sets of zero measure in R2. Also note that the absence of conditions on the function hallows us to consider the “zero dimensional” Furstenberg problem. However, this bound does not provide any substantial improvement, since the zero dimensionality property of the function his shared by the function h2. This is because the proof above, in the case of the Fα-sets, gives the worse bound (dim(E)≥2α) when the parameter αis in (0,1 2). 4. The h→h√·bound In this section we will turn our attention to those functions hthat satisfy the bound h(x).xαfor α≤1 2. For these functions we are able to improve on the previously obtained bounds. We need to impose some growth conditions on the dimension function h. This conditions can be thought of as imposing a lower bound on the dimensionality of hto keep it away from the zero dimensional case. Remark 4.1. Throughout this section, the expected dimension function should be about h√·. We therefore need a step down from this function. For this, we will look at the gap ∆0(x) = h(x) h(x). The next lemma says that we can split the h-dimensional mass of a set E contained in an interval Iinto two sets that are positively separated. Lemma 4.2. 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. 16 URSULA MOLTER AND EZEQUIEL RELA with some suitable modifications of the construction made in [18], Remark 1.5, (p. 10). There, for each 0 < α ≤1, an Fα-set is constructed whose dimension is not greater than 1 2+3 2α. It is straightforward to modify that construction for it to hold even at the endpoint α= 0. We also include the following example of an F2 0-set Gof dimension zero. It will be constructed using the next result, which is Example 7.8 (p. 104) in [4]. In that example, Falconer constructs sets E, F ⊆[0,1] with dim(E) = dim(F) = 0 and such that [0,1] ⊆E+F. E×{1} −F×{0} x −y θ Figure 3. Consider G=E×{1}∪−F×{0}. This set Ghas clearly dimension 0, and contains two points in every direction θ∈[0; π 4]. For, if θ∈[0; π 4], let c= tan(θ), so c∈[0,1]. By the choice of Eand F, we can find x∈Eand y∈Fwith c=x+y. The points (−y, 0) and (x, 1) belong to Gand determine a segment in the direction θ(Figure 3 on page 16). 6. Acknowledgments We would like to thank to Michael T. Lacey for fruitful conversations during his visit to the Department of Mathematics at the University of Buenos Aires. 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Math., 88:35–39, 2002. Dedicated to the memory of Tom Wolff. [20] Thomas H. Wolff. Lectures on harmonic analysis, volume 29 of University Lecture Series. American Mathematical Society, Providence, RI, 2003. With a foreword by Charles Fefferman and preface by Izabella Laba, Edited by Laba and Carol Shubin. Departamento de Matem´ atica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pabell´ on I, 1428 Capital Federal, ARGENTINA, and CONICET, Argentina E-mail address, Ursula Molter: umolt[email protected].ar E-mail address, Ezequiel Rela: ere[email protected].ar