arXiv:1006.4862v3 [math.CA] 12 Nov 2012 OIE PIP368. CONICET Jarn´ık’s theorems. a that functions exten ([Hau18]). i.e., been functions functions, has power inequality dimension this general of more side of hand left the [MR10] In F ftype of 1 max (1) then ie ls.W ei ihtedfiiino lsia Furst classical of Furst definition 1.1. small the Definition very with construct begin We to class. able given being a in interested are we ec,i ednt by denote we if Hence, segment fteset the of set α hsrsac sprilyspotdb rns PICT2006-0 Grants: by supported partially is research This phrases. and words Key esuydmninpoete fst fFrtnegtp.I type. Furstenberg of sets of properties dimension study We 1991 Date ti nw [o9] K0] htdim that [KT01]) ([Wol99], known is It -set E eog oteclass the to belongs ue2,2018. 29, June : E ahmtc ujc Classification. Subject Mathematics otezr iesoa functions dimensional zero the to h value the > γ than Abstract. h asoffdmnino any of dimension Hausdorff the type segment anrsl st banasapdmninetmt o whol sets a these for estimate of dimension size sharp the a obtain zero on to estimates is result give main to measures Hausdorff set α ℓ ⊆ e or E E dmninlFrtnegtp es aey for Namely, sets. type Furstenberg -dimensional ,w osrc set a construct we 0, ntedrcinof direction the in R α 1 2 ∩ ∩ ic napeiu okw hwdthat showed we work previous a in Since . F 2 ℓ or e α ℓ ℓ n hr r xmlsof examples are there and e e sgetrta reulto equal or than greater is sti o ahdirection each for if -set F 2 1 ntedrcinof direction the in α For sgetrta reulto equal or than greater is α ssapfrtewoecaso usebr esassociated sets Furstenberg of class whole the for sharp is RUAMLE N ZQILRELA EZEQUIEL AND MOLTER URSULA For ML USEBR SETS FURSTENBERG SMALL sti o ahdirection each for if -set + Φ( α α 1 2 n(0 in F 2 ; usebr es asoffdmnin ieso function dimension dimension, Hausdorff sets, Furstenberg n(0 in α α inf = ) F , α ] subset a 1], 1. , . E ≤ ] subset a 1], e γ Introduction Φ( e uhta h asoffdmnin(dim dimension Hausdorff the that such ∈ { uhta h asoffdmnino the of dimension Hausdorff the that such dim F E F h α h γ ∈ 1 γ ) rmr 87,28A80. 28A78, Primary H fHudr ieso o greater not dimension Hausdorff of . F E α ≤ h e nti ae euegeneralized use we paper this In . ( γ of E F nteui iceteei line a is there circle unit the in e ihtepeetconstruction, present the with , 1 2 α E : ) H R nteui iceteei line a is there circle unit the in + -sets 2 ( of E E α scalled is 2 3 ewl losyta such that say also will We . ) ,α α, ∈ R E ≥ 2 2 1 F ihdim with salwrbudfor bound lower a is scalled is max α h } usebr set Furstenberg 17 BCTX4 and X149 UBACyT 0177, γ , ( x { log = ) ∈ 2 ,α α, entnecessarily not re (0 H usebr set Furstenberg ls of class e negsets. enberg e otecase the to ded , negst in sets enberg ( 1] − Our . + E γ particular n . ) ( 2 1 x 1 ≤ } of ), o any for 2 1 + 3 2 H α ) . , 2 URSULAMOLTERANDEZEQUIELRELA
1.1. Dimension Functions and Hausdorff measures. Definition 1.2. A function h will be called dimension function if it belongs to the class H which is defined as follows: H := h : [0, ) [0 : ), non-decreasing, right continuous, h(0) = 0 . { ∞ → ∞ } The important subclass of those h H that satisfy a doubling condition will ∈ be denoted by Hd:
Hd := h H : h(2x) Ch(x) for some C > 0 . { ∈ ≤ } Let g, h be two dimension functions. We will say that g is dimensionally smaller than h and write g h if and only if ≺ h(x) lim = 0. x 0+ g(x) → A function h H will be called a “zero dimensional dimension function” α ∈ if h x for any α> 0. We denote by H0 the subclass of those functions. As≺ usual, the h-dimensional (outer) Hausdorff measure h will be defined as follows. For a set E Rn and δ > 0, write H ⊆ h ∞ (E) = inf h(diam(Ei)) : E Ei, diam(Ei) < δ . Hδ ⊂ ( i i ) X [ The h-dimensional Hausdorff measure h of E is defined by H h h (E) = sup δ (E). H δ>0 H Recall that the Hausdorff dimension of a set E Rn is the unique real number s characterized by the following properties:⊆ r(E)=+ for all r
1+3α θ 1 1 + 3α (3) g(x)= x 2 log− , θ> . x 2 However, this modification will not be sufficient for our main objective, which is to reach the zero dimensional case. More precisely, we will focus at the endpoint α = 0, and give a complete answer about the exact dimension of a class of Furstenberg sets. We will prove in Theorem 3.3 that, for any 2 given γ > 0, there exists a set Eγ R such that ⊆ 1 1 (4) Eγ Fhγ for hγ (x)= γ 1 and dimH (Eγ ) . ∈ log ( x ) ≤ 2 This result, together with the results from [MR10] mentioned above, shows 1 that 2 is sharp for the class Fhγ . In fact, for this family both inequalities in 1 (1) are in fact the equality Φ(Fhγ )= 2 . In order to be able to obtain (4), it is not enough to simply “refine” the construction of Wolff. He achieves the desired set by choosing a specific set as the fiber in each direction. This set is known to have the correct dimension. To be able to reach the zero dimensional case, we need to handle the delicate 4 URSULAMOLTERANDEZEQUIELRELA issue of choosing an analogue zero dimensional set on each fiber. The main difficulty lies in being able to handle simultaneously Wolff’s construction and the proof of the fact that the fiber satisfies the stronger condition of having positive measure for the correct dimension function. The paper is organized as follows. In Section 2, for a given α > 0, we carefully develop the main construction of small Furstenberg α sets, and obtain dimension estimates for the class Fα, α > 0. In Section 3 we show that we can modify the argument of the previous section to include the zero-dimensional functions hγ defined in (4). The key ingredient is a lemma on Diophantine approximation regarding the size of zero dimensional fibers proved in Section 4. As usual, we will use the notation A . B to indicate that there is a constant C > 0 such that A CB, where the constant is independent of A and B. By A B we mean≤ that both A . B and B . A hold. ∼
2. Upper Bounds for Furstenberg-type Sets In this section we will concentrate on the right hand side of (1). This inequality has been proved by showing that there exists a set E in Fα such that s(E) = 0 for any s > 1+3α . However, by the result of Besicovitch H 2 cited in the Introduction, this does not necessarily imply that h(E) = 0 1+3α H for any h x 2 . Here we refine the arguments of Wolff to show, in ≻ 1+3α Theorem 2.4, that if θ > 2 there exists a set E in Fα such that for 1+3α 2 θ 1 hθ hθ := x log− x we have that (E) = 0. The purpose of this section is twofold.H First, we will carefully re-trace Wolff’s arguments to show what is the key to be able to sharpen the loga- rithmic gap to obtain (3) instead of (2). Second, and more importantly, we analyze the proof in detail to understand the nontrivial modification to be performed in Section 3. We begin with a preliminary lemma about a very well distributed (mod 1) sequence. Lemma 2.1. For n N and any real number x [0, 1], there is a pair of natural numbers 0 ∈j, k n 1, such that ∈ ≤ ≤ − k j log(n) x √2 . − n − n ≤ n2
This lemma is a consequence of Theorem 3.4 of [KN74], p125, in which an estimate is given about the discrepancy of the fractional part of the sequence nα n N where α is an irrational of a certain type. { We} ∈ also need to introduce the notion of G-sets, a common ingredient in the construction of Kakeya and Furstenberg sets. Definition 2.2. A G-set is a compact set E R2 which is contained in the strip (x,y) R2 : 0 x 1 such that for⊆ any m [0, 1] there is a line segment{ contained∈ in E≤ connecting≤ } x = 0 with x = 1∈ of slope m, i.e. m [0, 1] b R : mx + b E, x [0, 1]. ∀ ∈ ∃ ∈ ∈ ∀ ∈ Finally we need some notation for a thickened line. SMALL FURSTENBERG SETS 5
Definition 2.3. Given a line segment ℓ(x)= mx + b, we define the δ-tube associated to ℓ as Sδ := (x,y) R2 : 0 x 1; y (mx + b) δ . ℓ { ∈ ≤ ≤ | − |≤ } Now we are ready to prove the main result of this section.
1+3α 2 θ 1 Theorem 2.4. For α (0, 1] and θ > 0, define hθ(x) = x log− ( x ). 1+3α ∈ h Then, if θ > , there exists a set E Fα with θ (E) = 0. 2 ∈ H j Proof. Fix n N and let nj be a sequence such that nj >n . We consider ∈ +1 j T to be the set defined as follows: 1 3 p 1 T = x , : j p,q ; q nα; x < . ∈ 4 4 ∀ ∃ ≤ j | − q | n2 ( j )
It can be seen that dimH (T )= α (see Section 4, Theorem 4.8). 1 t 1 1 3 1 3 If ϕ(t) = − and D = ϕ [ , ] , we have that ϕ : D [ , ] is t√2 − 4 4 → 4 4 bi-Lipschitz. Therefore the set 1 t 1 T ′ = t R : − T = ϕ− (T ) ∈ t√2 ∈ also has Hausdorff dimension α. The main idea of our proof is to construct a set for which we have, essen- tially, a copy of T ′ in each direction and simultaneously keep some optimal covering property. Define, for each n N, ∈ p 1 3 α Γn := , ,q n q ∈ 4 4 ≤ and 1 t p 1 Qn = t : − = Γn = ϕ− (Γn). √ t q ∈ 2 To count the elements of Γn (and Qn), we take into account that
nα ⌊ ⌋ 1 j nα ( nα + 1) . nα 2 n2α. ≤ 2⌊ ⌋ ⌊ ⌋ ⌊ ⌋ ≤ j X=1 2α Therefore, #(Qn) . n . For 0 j, k n 1, define the line segments ≤ ≤ − j k ℓjk(x) := (1 x) + x√2 for x [0, 1], − n n ∈
δn log(n) and their δ -tubes S with δ = 2 . We will use during the proof the n ℓjk n n notation Sn instead of Sδn . Also define jk ℓjk
n (5) Gn := Sjk. [jk Note that, by Lemma 2.1, all the Gn are G-sets. 6 URSULAMOLTERANDEZEQUIELRELA
For each t Qn, we look at the points ℓjk(t), and define the set S(t) := n ∈ 2 ℓjk(t) . Clearly, #(S(t)) n . But if we note that, if t Qn, then { }j,k=1 ≤ ∈ ℓjk(t) 1 t j k p j k pj + kq 0 = − + = + = < 2, ≤ t√2 t√2 n n q n n nq we can bound #(S(t)) by the number of non-negative rationals smaller than 2 of denominator qn. Since q nα, we have #(S(t)) n1+α. Considering ≤ ≤ all Q S t . n1+3α the elements of n, we obtain # t Qn ( ) . Let us define ∈ S √2 Λn := (x,y) Gn : x t 2 for some t Qn . ( ∈ | − |≤ n ∈ )
log(n) Claim 2.5. For each n, take δn = n2 . Then Λn can be covered by Ln 1+3α balls of radius δn with Ln . n . To see this, it suffices to set a parallelogram on each point of S(t) for each 2 t in Qn. The lengths of the sides of the parallelogram are of order n− and log(n) log(n) n2 , so their diameter is bounded by a constant time n2 , which proves the claim. Back to the proof of the Theorem: We can now begin with the recursive construction that leads to the desired set. The starting point F0 will be essentially any G-set. We only add the assumption that it can be written as
M0 δ0 F0 = S 0 , ℓi i [=1 0 0 0 (the union of M0 line segments ℓi = mi + bi with appropriate orientation 0 and δ -thickened). Each Fj to be constructed will be a G-set of the form
Mj δj j j j Fj := S j , with ℓi = mi + bi . ℓi i [=1 Having constructed Fj, consider the Mj affine mappings
j δj Ai : [0, 1] [ 1, 1] S j 1 i Mj, × − → ℓi ≤ ≤ defined by x 1 0 x 0 Aj = + . i y mj δj y bj i i Here is the key step: by the definition of T , we can choose the sequence nj to grow as fast as we need (this will not be the case in the next section). For example, we can choose nj+1 large enough to satisfy
(6) log log(nj+1) > Mj j and apply Ai to the sets Gnj+1 defined in (5) to obtain
Mj j Fj+1 = Ai (Gnj+1 ). i [=1 SMALL FURSTENBERG SETS 7
Since Gnj+1 is a union of thickened line segments, we have that
Mj+1 δj+1 Fj+1 = S j+1 , ℓi i [=1 j+1 for an appropriate choice of Mj+1, δj+1 and Mj+1 line segments ℓi . From j the definition of the mappings Ai and since the set Gnj+1 is a G-set, we conclude that Fj+1 is also a G-set. Define
Ej := (x,y) Fj : x T ′ . { ∈ ∈ } To cover Ej, we note that if (x,y) Ej, then x T ′, and therefore there p ∈ ∈ exists a rational Γn with q ∈ j 1 1 x p x r > − = ϕ(x) ϕ(r) | − |, for some r Qn . n2 √ − q | − |≥ √ ∈ j j x 2 2
Mj−1 j 1 Therefore (x,y) A − (Λn ), so we conclude that Ej can be covered ∈ i=1 i j 1+3α log(nj ) by Mj 1nj balls of diameter at most 2 . By our choice of nj, we − S nj 1+3α obtain that Ej admits a covering by log log(nj)nj balls of the same di- ameter. Therefore, if we set F = Fj and E := (x,y) F : x T we j { ∈ ∈ ′} obtain that T log(n ) hθ (E) . n1+3α (log log(n )) h j δj j j θ 2 H nj ! 1+3α 2 2 1+3α log(nj) θ nj . nj (log log(nj)) 2 log− nj ! log(nj)! 1+3α θ 1+3α +ε θ . (log log(nj)) log(nj) 2 − . log 2 − (nj) 1+3α for large enough j. Therefore, for any θ> 2 , the last expression goes to zero. In addition, F is a G-set, so it must contain a line segment in each direction m [0, 1]. If ℓ is such a line segment, then ∈ dimH (ℓ E) = dimH (T ′) α. ∩ ≥ The final set of the proposition is obtained by taking eight copies of E, rotated to achieve all the directions in S.
3. Upper Bounds for small Furstenberg-type Sets
In this section we will focus on the class Fα at the endpoint α = 0. Note that all preceding results involved only the case for which α> 0. Introducing the generalized Hausdorff measures, we are able to handle an important class of Furstenberg type sets in F0. The idea is to follow the proof of Theorem 2.4. But in order to do that, we need to replace the set T by a generalized version of it. A na¨ıve approach would be to replace the α power in the definition of T by a slower increasing function, like a logarithm. But in this case it is not clear that the set T fulfills the condition of having positive measure for the corresponding dimension function (recall that we want to construct a set in Fhγ ). More precisely, we will need the following lemma. 8 URSULAMOLTERANDEZEQUIELRELA
Lemma 3.1. Let r > 1 and consider the sequence n = nj defined by 4 { } 1 r j r n − 2 nj = e 2 j 1 , the function f(x) = 2 log(x ) 2 and the set
1 3 p 1 T = x , Q : j p,q ; q f(nj); x < . ∈ 4 4 \ ∀ ∃ ≤ | − q | n2 ( j ) h 1 Then we have that (T ) > 0 for h(x)= 1 . H log( x ) Remark 3.2. We postpone the proof of this lemma to the next section, but we emphasize the following fact: in this case, the construction of this new set T , does not allow us, as in (6), to freely choose the sequence nj. On one hand we need the sequence to be quickly increasing to prove that the desired set is small enough, but not arbitrarily fast, since on the other hand, we need to impose some control to be able to prove that the fiber has the appropriate positive measure. With this lemma, we are able to prove the main result of this section. We have the next theorem. 1 Theorem 3.3. Let h = 1 . There exists a set E Fh such that log( x ) ∈ 1 dimH (E) . ≤ 2 Proof. We will use essentially a copy of T in each direction in the construc- tion of the desired set to fulfill the conditions required to be an Fh-set. Let T be the set defined in Lemma 3.1. Define T ′ as
1 t 1 T ′ = t R : − T = ϕ− (T ), ∈ t√2 ∈ where ϕ is the same bi-Lipschitz function from the proof of Theorem 2.4. h Then T ′ has positive -measure. Let us define the corresponding sets of Theorem 2.4 for this generalizedH case. p 1 3 Γn := , ,q f(n) , q ∈ 4 4 ≤
1 t p 1 Qn = t : − = Γn = ϕ− (Γn). √ t q ∈ 2 2 r 2 r Now the estimate is #(Qn) . f (n) = log (n ) log (n), since ∼ f(n) ⌊ ⌋ 1 j f(n) ( f(n) + 1) . f(n) 2. ≤ 2⌊ ⌋ ⌊ ⌋ ⌊ ⌋ j X=1 n For each t Qn, define S(t) := ℓjk(t) . If t Qn, following the ∈ { }j,k=1 ∈ previous ideas, we obtain that r #(S(t)) . n log 2 (n), and therefore
3r # S(t) . n log(n) 2 . t Qn ∈[ SMALL FURSTENBERG SETS 9
Now we estimate the size of a covering of √2 Λn := (x,y) Gn : x t 2 for some t Qn . ( ∈ | − |≤ n ∈ ) log(n) For each n, take δn = n2 . As before, the set Λn can be covered with Ln 3r balls of radius δn with Ln . n log(n) 2 . Once again, define Fj, F , Ej and E as before. Now the sets Fj can be 3r log(n ) 2 j covered by fewer than Mj 1nj log(nj) balls of diameter at most 2 . − nj 2 Now we can verify that, since each Gn consist of n tubes, we have that 2 2 Mj = M n n . 0 1 · · · j Now we are again at the key point: by the choice of the sequence nj in the definition of T , we can also verify that the sequence satisfies the relation{ } 2 2 log nj Mj = M n n , and therefore we have the bound +1 ≥ 0 1 · · · j 3r log log(nj)nj log(nj) 2 1 dimH (E) dimB(E) lim = , ≤ ≤ j 2 1 2 log nj log− (nj) where dimB stands for the lower box dimension. Finally, for any m [0, 1] we have a line segment ℓ with slope m contained in F . It follows∈ that h(ℓ E)= h(T ) > 0. H ∩ H ′ We remark that the argument in this particular result is essentially the same needed to obtain the family of Furstenberg sets Eγ Fhγ for hγ(x)= 1 1 ∈ γ 1 , γ R+, such that dimH (Eγ ) 2 announced in the Introduction. log ( x ) ∈ ≤ 4. Proof of Lemma 3.1 The purpose of this section is to prove Lemma 3.1. Our proof relies on a variation of a Jarn´ık type theorem on Diophantine approximation. We begin with some preliminary results on Cantor type constructions that will be needed. 4.1. Cantor sets. In this section we introduce the construction of sets of Cantor type in the spirit of [Fal03]. By studying two quantities, the number of children of a typical interval and some separation property, we obtain sufficient conditions on these quantities that imply the positivity of the h- dimensional measure for a test function h H. We will need a preliminary elemental lemma∈ about concave functions. The proof is straightforward. Lemma 4.1. Let h H be a concave dimension function. Then ∈ a min a, b h(b) for any a, b R . { }≤ h(a) ∈ +
x+y h(x)+h(y) Proof. Since h is concave, we have h 2 2 . We consider two separate cases: ≤ If b a then a h(b) a h(a)= a = min a, b . • ≥ h(a) ≥ h(a) { } If a > b, then a h(b) b h(b)= b = min a, b by concavity of h. • h(a) ≥ h(b) { } 10 URSULAMOLTERANDEZEQUIELRELA
The following lemma is a natural extension of the “Mass Distribution Principle” to the dimension function setting. Lemma 4.2 (h-dimensional mass distribution principle). Let E Rn be a set, h H and µ a probability measure on E. Let ε > 0 and c⊆ > 0 be positive constants∈ such that for any U Rn with diam(U) < ε we have ⊆ µ(U) ch(diam(U)). ≤ Then h(E) > 0. H Proof. For any δ-covering we have 0 <µ(E) µ(U) c h(diam(U)). ≤ ≤ i i X X Then h > µ(E) and therefore h(E) > 0. Hδ c H Now we present the construction of a Cantor-type set (see Example 4.6 in [Fal03]).
Lemma 4.3. Let Ek be a decreasing sequence of closed subsets of the { } unit interval. Set E0 = [0, 1] and suppose that the following conditions are satisfied: k (1) Each Ek is a finite union of closed intervals Ij . (2) Each level k 1 interval contains at least mk intervals of level k. We will refer− to these as the “children” of an interval. (3) The gaps between the intervals of level k are at least of size εk, with 0 < εk+1 < εk.
Let E = k Ek. Define, for a concave dimension function h H, the quantity ∈ T h Dk := m1 m2 mk 1h(εkmk). · · · · − If lim Dh > 0, then h(E) > 0. k k H Proof. The idea is to use the version of the mass distribution principle from Lemma 4.2. Clearly we can assume that the property (2) of Lemma 4.3 holds for exactly mk intervals. So we can define a mass distribution on E assigning a mass of 1 to each of the m m intervals of level k. Now, m1 m 1 k ··· k · · · for any interval U with 0 < U < ε1, take k such that εk < U < εk 1. We will estimate the number of| intervals| of level k that could have| | non-empty− intersection with U. For that, we note the following: k 1 U intersects at most one Ij − , since U < εk 1. Therefore it can • k 1 | | − intersect at most mk children of Ij − . Suppose now that U intersects L intervals of level k. Then it must • U contain (L 1) gaps of size at least εk. Therefore, L 1 | | . εk − U U − ≤ Consequently U intersects at most | | + 1 2 | | intervals of level | | εk ≤ εk k. From these two observations, we conclude that 1 2 U 1 µ(U) min mk, | | = min εkmk, 2 U . ≤ m mk εk m mkεk { | |} 1 · · · 1 · · · SMALL FURSTENBERG SETS 11
Now, by the concavity of h, we obtain εkmk min εkmk, 2 U h(2 U ). { | |} ≤ h(εkmk) | | In addition (also by concavity), h is doubling, so h(2 U ) . h( U ) and then | | | | εkmkh( U ) h( U ) h( U ) µ(U) . | | = | | = | h| . m1 mkεkh(εkmk) m1 mk 1h(εkmk) D · · · · · · − k Dh > k 1 C k k Finally, if limk k 0, there exists 0 such Dh for 0 and we can k ≤ ≥ use the mass distribution principle with C and ε = εk0 . Remark 4.4. In the particular case of h(x)= xs, s (0, 1) we recover the result of [Fal03], where the parameter s can be expressed∈ in terms of the sequences mk and εk. For the set constructed in Lemma 4.3, we have
log(m1 mk 1) (7) dimH (E) · · · − . ≥ log(mkεk) − 4.2. Diophantine approximation - Jarn´ık’s Theorem. The central problem in the theory of Diophantine approximation is, at its simplest level, to approximate irrational numbers by rationals. A classical theorem due to Jarn´ık in this area is the following (see [Fal86]), which provides a result on the size of the set of real numbers that are well approximable. As usual, d(x, Z) denotes the distance from x to the nearest integer. Theorem 4.5. For β 2, define the following set: ≥ 1 Bβ = x [0, 1] Q : d(xq, Z) < for infinitely many q Z . ∈ \ qβ 1 ∈ − 2 Then dimH (Bβ)= β . In fact, there are several results (see [Khi24], [Jar31]) regarding the more general problem of estimating the size of the set q (8) B := x [0, 1] Q : d(xq, Z) for infinitely many q N , g ∈ \ ≤ g(q) ∈ where g is any positive increasing function. We can therefore see Lemma 3.1 as a result on the size of a set of well approximable numbers. We will derive the proof of Lemma 3.1 from the following proposition, where we prove a lower bound estimate for the set Bg. The original idea is from [Fal03], Theorem 10.3, where the result for the classical case in the form presented above in Theorem 4.5 is proved. 1 h Precisely, for h(x) = −1 1 2 we will find conditions on h with h h (g ( x )) ≺ to ensure that h(B ) > 0. To simplify the notation of the proof, let H g h(x) H ∆(h, h)(x) = h(x) and recall that d is the set of all dimension functions that satisfy a doubling condition. Proposition 4.6. Let g be a positive, increasing function satisfying (9) g(x) x2 (x 1) ≫ ≫ and 1 1 1 (10) g− (ab) . g− (a)+ g− (b) for all a, b 1. ≥ 12 URSULAMOLTERANDEZEQUIELRELA
Define Bg as in (8) and let h Hd be a concave dimension function such that 1 . Consider∈ a sequence that satisfies: h h(x)= −1 1 2 nk ≺ (g ( x )) { } (A) nk 3g(2nk 1). ≥ − (B) log(nk) g(nk 1). ≤ − h(x) 1 If ∆(h, h)(x)= h(x) = −1 1 2 satisfies h(x)g ( x ) 1 (11) lim > 0, k 2 1 k 6 g (2nk 2)∆(h, h) − log(nk)g(2nk−1) then h(B ) > 0. H g Proof. Define q (12) Gq := x [0, 1] Q : d(xq, Z) . ∈ \ ≤ g(q) For each q N, Gq is the union of q 1 intervals (with no rational numbers) ∈ 1 − 1 of length 2g(q)− and two more intervals of length g(q)− at the endpoints of [0, 1]. Now define for each q 1 1 G′ := Gq , 1 . q ∩ g(q) − g(q) Now, for each n N consider two prime numbers p ,p such that n ∈ 1 2 ≤ p1 < p2 < 2n (these can always be chosen for large n, see (13)). We will G G r1 prove that p′ 1 and p′ 2 are disjoint and well separated. Note that if p1 and r2 are centers of two of the intervals belonging to G and G , we have p2 p′ 1 p′ 2 r r 1 1 1 2 = r p r p , p − p p p | 1 2 − 2 1|≥ 4n2 1 2 1 2 since r1p2 r2p1 = 0. Therefore, taking into account this separation between the centers− and6 the length of the intervals, we conclude that for x G ∈ p′ 1 and y G , ∈ p′ 2 1 2 1 x y ( since g(n) n2). | − |≥ 4n2 − g(n) ≥ 8n2 ≫ Let n be the set of all the prime numbers between m and n and define Pm
Hn := Gp′ . p 2n ∈P[n 2 Then Hn is the union of intervals of length at least g(2n) that are separated 1 by a distance of at least 8n2 . Now we observe the following: If n N, n p 2n and I is an interval 3 p I ∈ ≤ ≤ with I > n , then at least |3 | of the intervals of Gp′ are completely con- tained| on| I. To verify this last statement, cut I into three consecutive and p I congruent subintervals. Then, in the middle interval there are at least |3 | m points of the form p . All the intervals of Gp′ centered at these points are 2 I completely contained in I, since the length of each interval of Gp′ is g(p) < |3|. SMALL FURSTENBERG SETS 13
n In addition, by the Prime Number Theorem, we know that #( 1 ) n P ∼ log(n) , so we can find n0 such that n (13) #( 2n) for n n . Pn ≥ 2 log(n) ≥ 0 Hence, if I is an interval with I > 3 , then there are at least | | n p I n n2 I | | > | | 3 2 log(n) 6 log(n) intervals of Hn contained on I. Now we will construct a Cantor-type subset E of Bg and apply Lemma 4.3. Consider the sequence nk of the hypothesis of the proposition and let { } E0 = [0, 1]. Define Ek as the union of all the intervals of Hnk contained in 1 Ek 1. Then Ek is built up of intervals of length at least and separated − g(2nk) 1 1 3 by at least ε = 2 . Moreover, since , each interval of E k 8n g(2n −1) n k 1 k k ≥ k − contains at least 2 nk mk := 6 log(nk)g(2nk 1) − intervals of Ek. Now we can apply Lemma 4.3 to E = Ek. Recall that εk denotes the separation between k-level intervals and mk denotes the number of chil- T dren of each of them. Consider h h, h Hd, and recall the notation h(x) ≺ ∈ ∆(h, h)(x)= h(x) . Then h Dk = m1 m2 mk 1h(εkmk) · · · · − (k 2) 2 2 6− − n2 nk 1 1 = · · · − h . log(n2) log(nk 1)g(2n1) g(2nk 2) 48 log(nk)g(2nk 1) · · · − · · · − −
Now we note that nk log(nk) and, by hypothesis (A), we also have that ≥ nk g(2nk 1). In addition, h is doubling, therefore it follows that we can bound≥ the first− factor to obtain that k 2 h 6− nk 1 Dk & − , 2 1 1 2 g (2nk 2)∆(h, h) (g− (log(nk)) + 2nk 1) − log(nk)g(2nk−1) − since, by hypothesis (B), nk satisfies log(nk 1) g(nk 2) and g satisfies (10). Now, again by hypothesis (B), − ≤ −
h 1 1 Dk k 2 . ≥ 6 g (2nk 2) ∆(h, h) 1 − log(nk)g(2nk−1) Thus, if 1 1 lim k 2 > 0, k 6 g (2nk 2) ∆(h, h) 1 − log(nk)g(2nk−1) then h(E) > 0 and therefore h(B ) > 0. H H g The following example not only illustrates this last result, but also will be crucial in the proof of Lemma 3.1. 14 URSULAMOLTERANDEZEQUIELRELA
2 x r Example 4.7. Define gr(x)= e for r> 0 and consider the set Bgr . Then 1 hr(x)= r 1 will be an expected lower bound for the dimension function log ( x ) 1 for the set Bgr . Consider the family hθ(x) = θ 1 (0 < θ < r), which log ( x ) hr(x) θ r 1 satisfy hθ hr. In this context, = log − ( ). Define the sequence nk ≺ hθ(x) x as follows: 2 r kn − nk = e k 1 . Clearly the sequence is admissible, since
(A) nk 3gr(2nk 1), and ≥ − (B) log(nk) gr(nk 1). − ≤ h Then, for the quantity Dk defined in Proposition 4.6, we have the following estimate: 2 r−θ r r θ 2 r (log log(nk)+ nk 1) − nk 1 2 h r +1 Dk & 2 − − 2 , M = 2 . Mn r ≥ Mn r 6ke k−2 6ke k−2 Finally, for any ε> 0 and M > 0, nk satisfies, for large k, 2 2 r r ε εknk−2 (εk M)nk−2 nk 1 e e − − 2 = 2 = k 1, Mn r Mn r 6 ≥ 6ke k−2 6ke k−2 so we conclude that lim Dh > 0. Therefore the set E B constructed in k k ⊆ gr the proof above satisfies hθ (E) > 0 for all θ < r. H 4.3. Another Jarn´ıktype Theorem and proof of Lemma 3.1. For the proof of Lemma 3.1 we will need a different but essentially equivalent formulation of Jarn´ık’s theorem. We first recall that in the proof of Theorem 2.4 we used the following theorem. j Theorem 4.8. Let n = nj j be an increasing sequence with nj n for { } +1 ≥ j all j N. For 0 < α 1, if An is defined as ∈ ≤ α p 1 n Q α Aα = x [0, 1] : j p,q ; q nj ; x < 2 , ( ∈ \ ∀ ∃ ≤ | − q | nj ) n then dimH (Aα)= α. For the proofs of Theorem 4.8 and Theorem 4.5, we refer the reader to [Jar31], [Bes34], [Egg52], [Fal86], and [Fal03]. n Now we want to relate the sets Aα and Bβ and their generalized versions. n It is clear that for any α (0, 1], we have the inclusion Aα B 2 . For ∈ ⊂ α α (0, 1], if x An then for each j N there exists a rational pj with ∈ ∈ α ∈ qj α pj 2 2 q n such that x n However, since the dimension of Aα coincides with the one of B 2 , one can α expect that both sets have approximately comparable sizes. We introduce the following definition, which is the extended version of n the definition of the set Aα in Theorem 4.8. Definition 4.9. Let n = nj j be any increasing nonnegative sequence of { } integers. Let f be an increasing function defined on R+. Define the set p 1 n Q Af := x [0, 1] : j p,q ; q f(nj); x < 2 . ( ∈ \ ∀ ∃ ≤ | − q | nj ) The preceding observation about the inclusion Aα Bβ can be extended ⊂ to this general setting. For a given g as in the definition of Bg, define Γ (x)= g 1(x2). Then the same calculations show that An B . g − Γg ⊂ g We will need a converse relation between those sets, since we want to prove n a lower bound for the sets Af from the estimates provided in Proposition 4.6. Lemma 4.10. Let g and Bg be as in Proposition 4.6. Define Γg(x) = 1 2 g− (x ). Then, if m = mk is the sequence defining the set E in the proof of Proposition 4.6, then{ the} set E is contained in An , where n 2Γg n = k = 1 { } g(mk) 2 . { } Proof. Recall that in the proof of Proposition 4.6 we define the sets Gq′ as r 1 r 1 a union of intervals of the form I = ; + Q. The sets Hn q − g(q) q g(q) \ 2n were defined as Hn := 2n Gp′ , where n is the set of primes between n p n P and 2n. We can therefore∈P write S n Hn := Ij . Now, given a sequence m = mk , for[ each k, the set Ek is defined as the { } union of all the intervals of Hmk that belong to Ek 1, where E0 = [0, 1]. If − mk E = Ek, any x E is in Ek and therefore in some of the I . It follows ∈ j that there exists integers r and q, q 2mk such that T ≤ r 1 1 1 1 2 x < < = , q 2g− (n ). − q g(q) g(m ) n2 ≤ k k k Therefore E An . ⊂ 2Γg We remark that the above inclusion implies that any lower estimate on E An the size of would also be a lower estimate for 2Γg . We now conclude the proof of Theorem 3.3 by proving Lemma 3.1: 1 Proof of Lemma 3.1: Let h(x)= 1 . For r> 1, consider the function gr, log( x ) the sequence m = mk and the set Er as in Example 4.7. Define f = 2Γgr , { } 4 r 1 r j n 2 2 2 nj−1 n and Af as in Lemma 4.10. It follows that f(x) = 2 log(x ) , nj = e and p 1 n Q Af := x [0, 1] : j p,q ; q f(nj); x < 2 . ( ∈ \ ∀ ∃ ≤ | − q | nj ) 16 URSULAMOLTERANDEZEQUIELRELA Note that Lemma 4.10 says that the inclusion E An always holds, for ⊂ 2Γg any defining function g, where E is the substantial portion of the set Bg (see Proposition 4.6). But we need the positivity of h(E) to conclude that the set An also has positive h-measure. For theH precise choices of h and g, 2Γg H we obtain this last property from Example 4.7. Precisely, h(E) > 0 and therefore the set An has positive h-measure. This concludesH the proof of f H Lemma 3.1 and therefore the set constructed in the proof of Theorem 3.3 fulfills the condition of being an Fh-set. As a final remark, we mention that it would be interesting to obtain sharp estimates in terms of generalized Hausdorff measures for any Fh class, not only for Fα, in the spirit of Theorem 2.4. The intuition here says that the reasonable dimension function for an upper bound for the class Fh is 3 √ h 2 . Therefore, a nice problem would be to construct an F set of zero · h h-measure for a dimension function h which is “very close” to h. And, further,H to estimate (as in Theorem 2.4) how close to h such an h can be in order to still allow us to perform the construction. Acknowledgments We would like to thank to Pablo Shmerkin for many interesting sugges- tions and remarks about this problem. References [Bes34] A. S. Besicovitch, Sets of fractional dimensions IV: On rational approximation to real numbers., J. London Math. Soc. 9 (1934), 126–131. [Egg52] H. G. Eggleston, Sets of fractional dimensions which occur in some problems of number theory, Proc. London Math. Soc. (2) 54 (1952), 42–93. [EK06] M´arton Elekes and Tam´as Keleti, Borel sets which are null or non-σ-finite for every translation invariant measure, Adv. Math. 201 (2006), no. 1, 102–115. [Fal86] K. Falconer, The geometry of fractal sets, Cambridge Tracts in Mathematics, vol. 85, Cambridge University Press, Cambridge, 1986. [Fal03] Kenneth Falconer, Fractal geometry, second ed., John Wiley & Sons Inc., Hobo- ken, NJ, 2003, Mathematical foundations and applications. [Hau18] Felix Hausdorff, Dimension und ¨außeres Maß, Math. Ann. 79 (1918), no. 1-2, 157–179. [Jar31] Vojtˇech Jarn´ık, Uber¨ die simultanen diophantischen Approximationen, Math. Z. 33 (1931), no. 1, 505–543. [Khi24] A. Khintchine, Einige S¨atze ¨uber Kettenbr¨uche, mit Anwendungen auf die Theorie der Diophantischen Approximationen, Math. Ann. 92 (1924), no. 1-2, 115–125. [KN74] L. Kuipers and H. Niederreiter, Uniform distribution of sequences, Wiley- Interscience [John Wiley & Sons], New York, 1974, Pure and Applied Mathe- matics. [KT01] Nets Katz and Terence Tao, Some connections between Falconer’s distance set conjecture and sets of Furstenburg type, New York J. Math. 7 (2001), 149–187 (electronic). [MR10] Ursula Molter and Ezequiel Rela, Improving dimension estimates for Fursten- berg-type sets, Adv. Math. 223 (2010), no. 2, 672–688. [OR06] L. Olsen and Dave L. Renfro, On the exact Hausdorff dimension of the set of Liouville numbers. II, Manuscripta Math. 119 (2006), no. 2, 217–224. [Rog98] C. A. Rogers, Hausdorff measures, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1998, Reprint of the 1970 original, With a foreword by K. J. Falconer. SMALL FURSTENBERG SETS 17 [Wol99] Thomas Wolff, Recent work connected with the Kakeya problem, Prospects in mathematics (Princeton, NJ, 1996), Amer. Math. Soc., Providence, RI, 1999, pp. 129–162. Departamento de Matematica,´ Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pabellon´ I, 1428 Capital Federal, ARGENTINA, and IMAS - CONICET, Argentina E-mail address, Ursula Molter: [email protected] E-mail address, Ezequiel Rela: [email protected]