PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 135, Number 10, October 2007, Pages 3151–3161 S 0002-9939(07)09019-3 Article electronically published on June 21, 2007

DIMENSION FUNCTIONS OF CANTOR SETS

IGNACIO GARCIA, URSULA MOLTER, AND ROBERTO SCOTTO

(Communicated by Michael T. Lacey)

Abstract. We estimate the packing measure of Cantor sets associated to non- increasing sequences through their decay. This result, dual to one obtained by Besicovitch and Taylor, allows us to characterize the functions recently found by Cabrelli et al for these sets.

1. Introduction A is a compact perfect and totally disconnected of the real line. In this article we consider Cantor sets of Lebesgue measure zero. Different kinds of these sets appear in many areas of , such as number theory and dynamical systems. They are also interesting in themselves as theoretical examples and counterexamples. A classical way to understand them quantitatively is through the Hausdorff measure and dimension. A function h :(0,λh] → (0, ∞], where λh > 0, is said to be a if it is continuous, nondecreasing and h(x) → 0asx → 0. We denote by D the set of dimension functions. Given E ⊂ R and h ∈ D,weseth(E)=h(|E|), where |E| is the of the set E. Recall that a δ-covering of a given set E is a countable family of of R covering E whose are less than δ.Theh-Hausdorff measure of E is defined as ∞ h (1.1) H (E) = lim h(Ui):{Ui} is a δ-covering of E . → δ 0 i=1 We say that E is an h-set if 0 < Hh(E) < +∞. s s g 0 When the dimension function is gs(x)=x ,fors ≥ 0, we set H := H s (H is the counting measure). The Hausdorff dimension of the set E, denoted by dim E, is the unique value t for which Hs(E)=0ifs>tand Hs(E)=+∞ if s

Received by the editors May 2, 2006. 2000 Mathematics Subject Classification. Primary 28A78, 28A80. Key words and phrases. Cantor sets, packing measure, Hausdorff dimension, dimension function.

c 2007 American Mathematical Society Reverts to public domain 28 years from publication 3151

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3152 I. GARCIA, U. MOLTER, AND R. SCOTTO

has Hausdorff dimension zero (see for example [Ols03]). Nevertheless, Cabrelli et al [CMMS04] showed that every Cantor set associated to a nonincreasing sequence a is dimensional; that is, they constructed a function ha ∈ D for which Ca is an −1/s ha-set. Moreover, they show that if the sequence a behaves like n ,thenha ≡ gs (see definition below) and therefore Ca is an s-set. But in other cases the behavior of these functions is not so clear. For example, there exists a sequence a such that Ca is an α-set but ha ≡ gα ([CHM02]). So these functions could be too general in order to give a satisfactory idea about the size of the set. To understand this situation we study the packing premeasure of these sets, which is defined as follows. A δ-packing of a given set E is a disjoint family of open balls centered at E with diameters less than δ.Theh-packing premeasure of E is defined as ∞ h { } P0 (E)=lim h(Bi): Bi i is a δ-packing of E . δ→0 i=1 As a consequence of our main result, which is dual to the one obtained in [BT54] and will be dealt with in Section 4, we are able to characterize completely when a dimension function is equivalent to a power function (Theorem 4.4). That is, for a nonincreasing sequence a and h ∈ D we obtain that ≡ ⇐⇒ Hs ≤ s ∞ h gs 0 < (Ca) P0 (Ca) < + .

Thus, to have that gα ≡ ha, it is not only necessary that Ca is an α-set but also α ∞ that P0 (Ca) < + .

2. Some remarks and definitions h By the definition of P0 , it is clear that it is monotone but it is not a measure h because it is not σ-additive; the h-packing measure P is obtained by a standard Ph { ∞ h } argument, (E)=inf i=1 P0 (Ei):E = i Ei . As with Hausdorff measures, given a set E there exists a critical value dimP E, s s the of E, such that P (E)=0ifs>dimP E and P (E)=+∞ { s} if s

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use DIMENSION FUNCTIONS OF CANTOR SETS 3153

Now we define a partial order in D, which will be our way of comparing the elements of D. Definition 2.1. Let f and h be in D. • f is smaller than h, denoted by f ≺ h,if lim h(x)/f(x)=0. x→0 • f is equivalent to h, denoted by f ≡ h,if h(x) h(x) 0

3. Cantor sets associated to nonincreasing sequences { } Definition 3.1. Let a = ak be a positive, nonincreasing and summable sequence. ∞ C Let Ia be a closed interval of length k=1 ak. We define ato be the family of all \ { } closed sets E contained in Ia that are of the form E = Ia j≥1 Uj,where Uj is a disjoint family of open intervals contained in Ia such that |Uk| = ak ∀k.

With this definition, every element of Ca has Lebesgue measure zero. From this family, we consider the Cantor set Ca associated to the sequence a constructed as follows: In the first step, we remove from Ia an open interval of length a1, resulting 1 1 in two closed intervals I1 and I2 . Having constructed the k-th step, we get the k k closed intervals I1 ,...,I2k contained in Ia. The next step consists in removing k k+1 from Ij an open interval of length a2k+j−1, obtaining the closed intervals I2j−1 and k k+1 ∞ 2 k I2j . Then we define Ca := k=1 j=1 Ij . Note that in this construction there is a unique form of removing open intervals at each step; also, note that for this construction it is not necessary for the sequence to be nonincreasing. {| k|} ≤ ≤ k Remark 3.2. Since a is nonincreasing, the sequence Ij (k,j),with1 j 2 and k ≥ 1, is (lexicographically) nonincreasing. We associate to the sequence a the summable sequences a and a defined as

an = a2r+1−1 and an = a2r , r ≤ r+1 ≤ ≤ ∀ with 2 n<2 ;thusan an an n.ThesetsCa and Ca are uniform, which means that for each k ≥ 1, the closed intervals of the k-thstephaveallthe

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3154 I. GARCIA, U. MOLTER, AND R. SCOTTO

same length. Observe that for each uniform Cantor set the Hausdorff and lower box dimensions coincide (cf. [CHM97]). To study the Hausdorff measure and dimension of these sets, Besicovitch and Taylor in [BT54] studied the decay of the sequence bn = rn/n,wherern = j≥n aj. They introduced the number

αn ∀ α(a) = lim αn, where nbn =1 n, n→∞

andshowedthatdimE ≤ α(a) for all E ∈ Ca.Infact,dimCa = α(a) (see, for example, [CMMS04]). Also, as a consequence of another result in [BT54] (see Proposition 4.1 below), { s ∞} (3.1) dim Ca =inf s>0 : lim nbn < + . In this paper (Theorem 4.2) we obtain the symmetrical result for the packing case: { s ∞} (3.2) ∆Ca =inf s>0:lim nbn < + .

On the other hand, the box dimensions of the sets in Ca are related to the constants log 1/k log 1/k (3.3) γ(a) = lim and β(a)= lim . →∞ k→∞ log ak k log ak In fact, by Propositions 3.6and3.7 of Falconer’s book [Fal97], the box dimension of E ∈ Ca exists if and only if γ(a)=β(a),andinthiscasedimB E = β(a). Moreover, as Falconer remarks, every E ∈ Ca has the same upper box dimension and the same lower box dimension. The former equals β(a), which in [Tri95] is shown to be equal to limn→∞ αn. In the following proposition we obtain the symmetrical result, that ≤ is, the latter is limn→∞ αn, but observe that γ(a) dimBCa and the inequality can be strict (cf. Propositions 3 and 5 in [CMMS04]).

Proposition 3.3. For every nonincreasing sequence a we have that dimBCa = dim Ca = α(a). s ∼ s Proof. First note that lim nbn lim nbn. In fact, to see the nontrivial inequality, − if {n } is a subsequence of the natural numbers let 2lj 1 ≤ n < 2lj ,so2lj bs ≤ j j 2lj 2n bs and therefore lim 2kbs ≤ 2 lim nbs .Nowfor2in ≤ n<2in+1, j nj k→∞ 2k n→∞ n ∞ ≤ in+k rn ak = 2 a2in+k k≥2in k=0 ∞ i +k−1 ≤ n ≤ − 2 2 a2in+k−1 =2r2in−1 2 r2in 1 ; k=0 s ≤ j s ≤ s hence limn→∞ nbn 4 limj→∞ 2 b2j 8 limn→∞ nbn. Then by (3.1) dim Ca =dimCa, and by Proposition 3.1of[CHM97]wehavethat ≤ dim Ca =dimBCa. On the other hand, N(ε, Ca) N(ε, Ca)(Ca can be mapped to Ca by a bijection which preserves the order; then, if two points of Ca are contained in an open set U, the corresponding points of Ca willbecontainedinanopenset ≤  of the same diameter as U), so dimBCa dimBCa. { s ∞} This proposition in particular asserts that dimBE =inf s>0 : lim nbn < + for any E ∈ Ca. On the other hand, since the critical exponent associated to

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use DIMENSION FUNCTIONS OF CANTOR SETS 3155

the packing premeasure of a given set is the upper box dimension, it is clear that { s ∞} dimBE =inf s>0:lim nbn < + . 4. Main results The next proposition is a result that generalizes the one established in [BT54] h for the functions gs to any function h ∈ D (cf. [CHM02]). It shows that H (Ca) behaves like nh(bn)whenn →∞. ∈ D 1 ≤Hh ≤ Proposition 4.1. For h , 4 limn→∞ nh(bn) (Ca) 4 limn→∞ nh(bn). 1 ≤Hh Proof. The lower inequality 4 limn→∞ nh(bn) (Ca) is obtained in exactly the same way as in [BT54] replacing gs by h. Hence, we only show here that Hh ≤ | k| ∞ i (Ca) 4 lim nh(bn). We begin by noting that Ij = i=0 2 a2k+i = b2k for 1 ≤ j ≤ 2k, k>0. Then, from the identities | k| I1 = a2k +(a2k+1 + a2k+1+1)+(a2k+2 + a2k+2+1 + a2k+2+2 + a2k+2+3)+... , | k−1| I1 = a2k−1 +(a2k+1−1 + a2k+1−1)+(a2k+2−1 + a2k+2−1 + a2k+2−1 + a2k+2−1)+... and Remark 3.2, we have the following estimate: | k|≤| k|≤| k−1| ≤ (4.1) Ij I1 I1 = b2k−1 b2k−1 , 2k k ≤ k − and hence j=1 h(Ij ) 2 h(b2k 1 ). Given δ>0thereexistskδ such that | k| ≥ ≥ I1 <δfor k kδ;thatis,fork kδ, the closed intervals of the k-th step Hh ≤ k form a δ-covering of Ca, which implies (Ca) 2 limk→∞ 2 h(b2k ), and therefore h H (Ca) ≤ 4 lim nh(bn).  Our main result is the following theorem, which is in some sense dual to the previous one. ∈ D 1 ≤ h ≤ Theorem 4.2. For any h , 8 limn→∞ nh(bn) P0 (Ca) 8 limn→∞ nh(bn).

Proof. For the first inequality, suppose that limn→∞ nh(bn) >d.Toprovethat h ≥ { } P0 (Ca) d/8, for each δ>0 it suffices to find a δ-packing Bi i of Ca with h(B ) >d/8. Observe that, since {a } is nonincreasing, then i i ⎛ n ⎞ ⎝ −k | k|⎠ ≤ k (4.2) h(b2k )=h 2 Ii h I1 . 1≤i≤2k { } By hypothesis there exists a subsequence nj j≥1 such that njh(bnj ) >d.Foreach k k +1 j,letkj be the unique integer for which 2 j ≤ nj < 2 j ;since{bn}n is decreasing, it follows from (4.2) that  kj +1 kj +1 kj k ≤ (4.3) d

| kj | Pick j large enough so that I1 <δ; since the diameter of this interval is smaller than the diameter of every interval of the kj − 1 step, the family of intervals k −2 − {B }2 j ,whereB is centered at the right endpoint of the interval Ikj 1 and |B | = i i=1 i 2i−1 i k − k | j | kj 2 j I1 , turns out to be a δ-packing of Ca,andby(4.3), i h(Bi)=2 h(I1 ) > d/8. { }N For the second inequality, if Bi i=1 is a δ-packing of Ca, we define { k ⊂ ≤ ≤ k} ki =min k : Ij Bi for some 1 j 2 .

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3156 I. GARCIA, U. MOLTER, AND R. SCOTTO

By the definition of ki, Bi is centered at a point of an interval of the ki − 1step, − − but it does not contain the interval, so |B | < |Iki 2|,whereIki 2 is the interval of i ji ji the ki − 2 step which contains the center of Bi. Then, by the monotony of h and (4.1), N N N ki−2 (4.4) h(B ) ≤ h(I ) ≤ h(b k −3 ). i ji 2 i i=1 i=1 i=1

Furthermore, we can assume that |B1|≥... ≥|BN |,sok1 ≥ ... ≥ kN .Let l1 > ... > lM denote those ki’s that do not repeat themselves. Let θm be the number of repeated lm’s; i.e., θm tells us how many of the Bi’s contain an interval { }N of the lm-th step but none of the previous ones. Since Bi i=1 is a disjoint family, l θ1 cannot be greater than the number of intervals of step l1,whichis21 ;each l −l ball of the packing associated to l1 contains 2 2 1 intervals of step l2 and therefore l2 l2−l1 θ2 ≤ 2 − θ12 . Continuing with this process we obtain   − − M1 M1 θ lM lM −li lM i θM ≤ 2 − θi2 =2 1 − , 2li i=1 i=1 M li ≤ and hence i=1 θi/2 1. l −3 Finally, choose δ sufficiently small such that 2 1 ≥ n0,where

sup nh(bn) ≤ lim nh(bn)+ε. n→∞ n≥n0 Then, N M θj l −3 ≤ j − ≤ h(Bi) − 2 h(b lj 3 ) 8( lim nh(bn)+ε) , 2lj 3 2 n→∞ i=1 j=1 and from this the theorem follows.  We are now ready to complete the characterization promised in the introduction. Note that the function ha ∈ D found in [CMMS04] is defined in such a way that ha ∞ ha(bn)=1/n for all n. Then, by Theorem 4.2 we obtain that P0 (Ca) < + , and therefore the Cantor sets associated to nonincreasing sequences not only are dimensional but also have a dimension function which simultaneously regularizes the covering and packing processes in the construction of the measures. We need the following lemma. ∈ D Hh ≤ h Lemma 4.3. Let h, g . In addition, assume that 0 < (Ca) P0 (Ca) < +∞.Then ≡ ⇐⇒ ≤ ∞ a) h g 0 < limn→∞ ng(bn) limn→∞ ng(bn) < + ; b) g ≺ h ⇐⇒ limn→∞ ng(bn)=+∞; c) h ≺ g ⇐⇒ limn→∞ ng(bn)=0. Hh h ∞ Proof. By Proposition 4.1 and Theorem 4.2, since (Ca) > 0andP0 (Ca) < + , there are constants 0

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use DIMENSION FUNCTIONS OF CANTOR SETS 3157

and g(yj ) ≥ −1 (2Ch) (nj +1)g(bnj +1), h(yj) from which the sufficient conditions also follow.  We have now the main theorem of this part. Theorem 4.4. Let a be a nonincreasing sequence, and let h ∈ D be such that h h 0 < H (Ca) ≤ P (Ca) < +∞.Then,forg ∈ D we have: ≡ ⇐⇒ Hg ≤ g ∞ a) h g 0 < (Ca) P0 (Ca) < + ; g b) g ≺ h ⇐⇒ H (Ca)=+∞; ≺ ⇐⇒ g c) h g P0 (Ca)=0. s Hs ≤ s ∞ In particular, h will be equivalent to x if and only if 0 < (Ca) P0 (Ca) < + . Proof. The proof is immediate from Proposition 4.1, Theorem 4.2 and Lemma 4.3. 

Corollary 4.5. Let α =dimCa and β =∆Ca.Takeha to be the dimension function of Ca. α a) If s<α and β 0,then ha gβ. b) In the case α<β, ha ≡ gs for no s ≥ 0. Moreover, if α 0, ha and gβ are not comparable.

α Proof. In the case H (Ca) < +∞,ifga were to satisfy gα ≺ ha, Proposition 2.2 h would imply that H a (Ca) = 0, which is a contradiction. Hence gα ⊀ ha.Analo- β ⊀ ha ∞ gously, P0 (Ca) > 0 implies that ha gβ,forifnotP0 (Ca)=+ , contradicting Theorem 4.2. The rest of the claims of item a) are immediate from Theorem 4.4. ≡ ≥ Hs ≤ s ∞ To show b), if ha gs for some s 0, then 0 < (Ca) P0 (Ca) < + ; therefore α = β. Moreover, by Proposition 2.2, it follows that gs ⊀ h when s> dim Ca,andalsoha ⊀ gs if 0 ≤ s<β. 

Note that, since α =dimBCa and β = dimBCa,partb) of this corollary empha- ≡ sizes that in order to have ha gs we need dimBCa = dimBCa. But this latter condition and the fact that Ca is an α-set are not sufficient to ensure the equiva- lence, and thus the hypothesis of Theorem 4.4 a) cannot be weakened to existence of box dimension and Ca being an α-set.

Example 4.6. For each 0

k 1 s+ε To construct this sequence we set λk = k ,where  2 s log l , 2m

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3158 I. GARCIA, U. MOLTER, AND R. SCOTTO

and, using (3.3), that dim Ca = dimBCa = s. Next we check the claims about the measures: j s i ≥ 1 s ∀ a) 0 < limn→∞ nbn:Sinceb2k = i≥0 2 λk+1+i and λj ( 2 ) j,wehave that ⎛ ⎞ ⎛ ⎞ s s   k+1+i −   1 − s i ( s 1)i k s k ⎝ 1 ⎠ 1 ⎝ 1 ⎠ 2 b k ≥ 2 = = c > 0 2 2 2 2 s i≥0 i≥0

and remember that lim nbs ∼ lim 2kbs . n→∞ n k→∞ 2k s 2j −1 i ∞ j b) limn→∞ nbn =+ :Letnj =2 , j odd. Then bnj = i≥0 2 λ2 +i > λ2j ; therefore,

j − j − 2 (j 1) log 2 −1 s 2 1 s 2j +(j−1) log 2 ≥ j njbnj 2 λ2 =2 , which increases to infinity with j. s ∞ 2j c) limn→∞ nbn < + :Nowwesetnj =2 for j odd and observe that ⎛ ⎞ ⎛ ⎞ s s 2i s ⎝ i ⎠ ⎝ 2i+l+2j ( 1−s )−1 ⎠ j s i (4.5) njbnj = nj 2 λ2 +1+i = 2 λ2 +l . i≥0 i≥j l=1 Each sum in l is bounded by a geometric term; more precisely, if j is sufficiently large there is a constant Cs depending only on s such that

i   2 i−j 2i+l+2j ( 1−s ) 1 2 s λ i ≤ C , 2 +l s 2 l=1 or equivalently,

i   i j 1−s − 2 2 +2 ( s )+i j l 1 (4.6) 2 λ i ≤ C . 2 +l s 2 l=1

i 2 l This is easy to check when i is odd. For i even we obtain l=1 2 λ2i+l < 2i 1 s+ε Cs 2 for small ε. Thus (4.6) will hold if     1 − (s + ε) 1 − s (4.7) 2i ≥ 2j + i − j. s + ε s But notice that (4.7) is true for all i>jchoosing ε sufficiently small and j large enough so that ε2j <ε. Remark 4.7. Proposition 4.1 and Theorem 4.2 are not valid in general as the fol- lowing arguments show. a ≤ a˜ If a is a nonincreasing sequence anda ˜ is any rearrangement of a,thenrn rn; hence, by (3.1) and since Ca˜ ∈ Ca, { a˜ s ∞} ≥ ≥ inf s>0 : lim n(bn) < + dim Ca dim Ca˜, and each positive and summable sequence a has a rearrangement z for which dim Cz = 0 (cf. [CMPS05]). This shows the counterexample for Proposition 4.1.

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use DIMENSION FUNCTIONS OF CANTOR SETS 3159

In the case of Theorem 4.2 let us see that (3.2) need not be true for a general { a˜ s sequence. Leta ˜ be any rearrangement of a.Letη(˜a):=inf s>0:lim n(bn) < +∞} and consider β(˜a)asdefinedinSection3.Notethatift>η(˜a), thena ˜n < 1−1/t ∀ ≤ t ≤ η(˜a) Cn n, and hence β(˜a) 1−t , which implies that β(˜a) 1−η(˜a) . Therefore β(˜a) ≤ 1+β(˜a) η(˜a). Thus, to show that this theorem fails in general, we exhibit a nonincreasing sequence a and a rearrangementa ˜ of it for which β(˜a) (4.8) β(a) < , 1+β(˜a)

and therefore ∆Ca˜ =∆Ca = β(a) <η(˜ a), where ∆Ca˜ is the critical exponent for s 1 p P0 , the packing premeasure. For an = n ,weset ⎧ k j ⎨ a3k ,n= log 3 ,n=3 ∀j, k   j∀ a˜n = ⎩ alog 3k,n=3 ,n= log 3 j, an, otherwise, where s denotes the smallest integer greater than s. Observe that this is a re- l arrangement of a since the sequence {log 3k} is strictly increasing and log 33  = 3j for each l and each j. Then it is easy to check that β(a)=1/p and that log 3 log 3 β(˜a)= p . Therefore (4.8) holds for p> log 3−1 . However these propositions keep on being true if we ask the Cantor set to be uniform with no further assumptions on the sequence. This can be seen in Theorem 4.2 since inequalities (4.3) and (4.4), where the decay assumption is needed, are still true for uniform Cantor sets. For Proposition 4.1 note that Lemma 4 of [BT54] does not need the decay. Moreover, to each Cantor set associated to a nonincreasing sequence corresponds a uniform Cantor set with equivalent h-Hausdorff measure and h-packing premeasure. Finally we give an application of the results of this section. q p Let p>1andq ∈ R. Consider the sequence a defined by an =(logn) /n , ∀ n>1. We denote by Cp,q the Cantor set associated to a and define the dimension 1 q function hp,q(x)=x p /(− log x) p . Since γ(a)=β(a)=1/p,wehavethatdimCp,q =dimB Cp,q =1/p for any q. Moreover, if q =0,Cp = Cp,0 is a 1/p-set (cf. [CMPS05]). Even more, as in this 1 ∼ p−1 p ∞ case rn 1/n , Theorem 4.2 implies that P0 (Ca) < + . The next corollary extends these results. h Hhp,q ≤ p,q ∞ Corollary 4.8. With the above notation, 0 < (Cp,q) P0 (Cp,q) < + . In particular, Cp,q is an hp,q-set.

q p−1 Proof. We show that rn ∼ (log n) /n and from this it is easy to see that hp,q(bn) ∼ 1/n. First suppose that q ≥ 0. In this case we have that (log n)q r ≥ (log n)q k−p ≥ c n 1 np−1 k≥n

and ∞ q ≤ (log t) rn p dt, n−1 t

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3160 I. GARCIA, U. MOLTER, AND R. SCOTTO

so integrating by parts we get   1 (log(n − 1))q ∞ (log t)q−1 r ≤ + q dt n p − 1 (n − 1)p−1 tp  n−1 (log n)q (log n)q−1 = c + O p np−1 np−1 (log n)q ≤ c , 2 np−1 where c1 and c2 are constants depending only on p and q. q p−1 Now suppose that q<0. In this case it is easy to see that rn ≤ c3(log n) /n . On the other hand, integrating by parts twice and since q(q − 1) > 0weobtain   1 (log n)q q (log n)q−1 q(q − 1) ∞ (log t)q−2 r ≥ + + dt n p − np−1 p − np−1 p − tp−1 1  1  1 n 1 (log n)q q ≥ 1+ (log n)−1 ; p − 1 np−1 p − 1 q p−1 taking n sufficiently large it follows that rn ≥ c4(log n) /n . 

Note that hp,q hs,t if and only if (1/p, q) ≤l (1/s, t)(≤l denote the lexicograph- 1 ,q 1 ,0 1 ical order in (0, 1)×R). Define H p = Hhp,q ,sothatH p = H p . As a consequence of the above result we can conclude the following. If q<0, (1/p, q) 0, p,q 1 then (1/p, 0) 0 implies that H p (Cp,q)=+∞. Hence, the family {hp,q} provides a more accurate classification than the usual {gs}, i.e., it distinguishes more sets; furthermore, the dimension induced by this family can be seen as the set ((0, 1) × R) ∪{(0, 0)}∪{(1, 0)} ordered with ≤l, and the set function dim turns out to be the restriction of this order to (0, 1) ×{0}.

Acknowledgments We thank C. Cabrelli and K. Hare for sharing their insight into this problem and giving us access to their unpublished notes [CHM02]. The authors are also grateful to the referee’s useful comments.

References

[Bes39] E. Best. A closed dimensionless linear set. Proc. Edinburgh Math. Soc. (2), 6:105–108, 1939. MR0001824 (1:302f) [BT54] A. S. Besicovitch and S. J. Taylor. On the complementary intervals of a linear closed set of zero Lebesgue measure. J. London Math. Soc., 29:449–459, 1954. MR0064849 (16:344d) [CHM02] Carlos Cabrelli, K. Hare, and Ursula M. Molter. Some counterexamples for Cantor sets. Unpublished Manuscript, Vanderbilt, 2002. [CHM97] Carlos A. Cabrelli, Kathryn E. Hare, and Ursula M. Molter. Sums of Cantor sets. Ergodic Theory Dynam. Systems, 17(6):1299–1313, 1997. MR1488319 (98k:28009) [CMMS04] Carlos Cabrelli, Franklin Mendivil, Ursula M. Molter, and Ronald Shonkwiler. On the Hausdorff h-measure of Cantor sets. Pacific J. Math., 217(1):45–59, 2004. MR2105765 (2005h:28013) [CMPS05] C. Cabrelli, U. Molter, V. Paulauskas, and R. Shonkwiler. Hausdorff measure of p- Cantor sets. Real Anal. Exchange, 30(2):413–433, 2004/05. MR2177411 (2006g:28012) [Fal97] Kenneth Falconer. Techniques in geometry. John Wiley & Sons Ltd., Chichester, 1997. MR1449135 (99f:28013)

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use DIMENSION FUNCTIONS OF CANTOR SETS 3161

[Ols03] L. Olsen. The exact Hausdorff dimension functions of some Cantor sets. Nonlinearity, 16(3):963–970, 2003. MR1975790 (2004g:28009) [Rog98] C. A. Rogers. Hausdorff measures. Cambridge Mathematical Library. Cambridge Uni- versity Press, Cambridge, 1998. MR1692618 (2000b:28009) [Tri82] Claude Tricot, Jr. Two definitions of fractional dimension. Math. Proc. Cambridge Philos. Soc., 91(1):57–74, 1982. MR633256 (84d:28013) [Tri95] Claude Tricot. Curves and . Springer-Verlag, New York, 1995. MR1302173 (95i:28005) [TT85] S. James Taylor and Claude Tricot. Packing measure, and its evaluation for a Brownian path. Trans. Amer. Math. Soc., 288(2):679–699, 1985. MR776398 (87a:28002)

Departamento de Matematica,´ Facultad de Ingenier´ıa Qu´ımica, Universidad Nacional del Litoral, Santa Fe, Argentina and IMAL CONICET UNL E-mail address: [email protected] Departamento de Matematica,´ Facultad de Ciencias Exactas y Naturales, Universi- dad de Buenos Aires, Ciudad Universitaria, Pabellon´ I, Capital Federal, Argentina and CONICET, Argentina E-mail address: [email protected] Departamento de Matematica,´ Universidad Nacional del Litoral, Santa Fe, Ar- gentina E-mail address: [email protected]

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use