Real Analysis Exchange Vol. 302), 2004/2005,. 20 C. Cabrelli, U. Molter, Deartamento de Matemática, Facultad de Cs. Exactas y Naturales, Universidad de Buenos Aires and CONICET, Pabellón I - Ciudad Universitaria, 428 - Buenor Aires, Argentina. email: cabrelli@dm.uba.ar and umolter@dm.uba.ar V. Paulauskas, Deartment of Mathematics and Informatics, Vilnius University, Lithuania and the Institute of Mathematics and Informatics, Vilnius, Lithuania. email: vaul@ieva.maf.vu.lt R. Shonkwiler, School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-060. email: shonkwiler@math.gatech.edu web: www.math.gatech.edu/ shenk HAUSDORFF MEASURE OF -CANTOR SETS Abstract In this aer we analyze Cantor tye sets constructed by the removal of oen intervals whose lengths are the terms of the -sequence, {k } k=. We rove that these Cantor sets are s-sets, by roviding shar estimates of their Hausdorff measure and dimension. Sets of similar structure arise when studying the set of extremal oints of the boundaries of the so-called random stable zonotoes. Introduction and Notation. By a general Cantor set, we mean a comact, erfect, totally disconnected subset of the real line. In this aer we will only consider Cantor sets of zero Lebesgue measure. Cantor sets arise in many different settings including as examles having many surrising roerties. They can be constructed in different ways. Key Words: Hausdorff measure, Hausdorff dimension, Cantor like sets Mathematical Reviews subject classification: 28A78 Received by the editors May, 2002 Communicated by: R. Daniel Mauldin Part of this research was carried out during a visit of Cabrelli, Molter and Paulauskas, at the School of Mathematics, Georgia Institute of Technology. These authors acknowledge the suort received during their visit. Cabrelli and Molter are also artially suorted by grants: CONICET, PID 456/98, UBACyT X058, X08 and BID-OC/AR PICT 0334, 5033.
2 C. Cabrelli, U. Molter, V. Paulauskas and R. Shonkwiler General Cantor sets can be constructed in a manner similar to that of the classical middle-third Cantor set. Let λ = {λ k } k N be a sequence of ositive numbers, such that k λ k = K < +. We associate to such a sequence, the Cantor set C λ in the following way. Start with the closed interval I 0 = [0, K] of length, I 0 = K = k= λ k. Clearly by normalization we can always achieve I 0 = [0, ].) In what follows, we use the notation I for the lengthi) = diami) of any interval. In the first ste, remove from I 0 an oen interval of length λ, obtaining the two closed intervals of ste, I0 and I, and a ga of length λ between them. Shortly we will see that the location of the ga to be removed is uniquely determined. Having comleted ste k, we will have 2 k closed intervals Il k, l = 0,..., 2k. From each of them we remove an oen interval of length equal to the next unused term of the sequence; thus, from Il k, we remove an oen interval of length λ 2k +l. Again, we will see that the location of the ga is uniquely determined. This forms in Il k, two closed sub-intervals Ik+ 2l and I k+ 2l+, with resect to which Il k = I k+ 2l + λ 2k +l + I k+ 2l+. As noted above, in order for this construction to be ossible, the osition of the gas removed at each ste is not arbitrary. Since the length of the interval I 0 equals the sum of the lengths of all the intervals removed in the construction, there is a unique way of doing this construction. That is, the length of each of the remaining intervals at ste k should be exactly the sum of the lengths of all the gas that will be removed from it later in the construction. So, the two remaining intervals of ste will have lengths: I 0 = n= 2 n j=0 λ 2 n +j, I = n λ 2 n +j. n= j=2 n In general, for all k = 0,,... and l = 0,,..., 2 k, we have I k l = n=k l+)2 n k j=l2 n k λ 2 n +j. ) We denote by C k λ the union of the closed intervals at the kth ste C k λ = 2 k l=0 I k l.
Hausdorff Measure of -Cantor Sets 3 Then C λ = Cλ. k k=0 We will say that the Cantor set constructed in this way using a sequence λ, is the Cantor set associated to the sequence λ. Remark. This construction is quite general since in fact any Cantor set of zero Lebesgue measure) can be obtained in this way for an aroriate choice of the sequence. Namely, let C be a Cantor set in R and let I 0 be the smallest interval containing C. The comlement of C in I 0 is a countable union of oen intervals U i, such that i N U i = I 0. The following rocedure will show how to define a sequence a = {a k }, such that C a = C. Let U i be a ga of maximal length and define a = U i. Next choose U i2 a ga of maximal length to the left of U i, and U i3 a ga of maximal length to the right of U i. Now define a 2 = U i2 and a 3 = U i3. In the next ste define a 4 through a 7 by icking a ga of maximal length in each of the remaining intervals i.e., I 0 \ U i U i2 U i3 ).) Continuing in this fashion, the sequence a = {a k } satisfies C a = C. It is clear from the construction of the Cantor set associated to a given sequence, that the secific order in which the gas aear in the sequence determines the resulting Cantor set. Of central imortance in this aer is the investigation of the effect on Hausdorff dimension defined below) due to rearrangements of the sequence of gas. Definition. Let σ : N N be a bijective ma; we say that the sequence {λ σk) } k N is a rearrangement of λ and denote it by σλ). Remark. In general, a rearrangement of the original sequence yields a different Cantor set. As we will see in this aer, the new Cantor set can have a different Hausdorff dimension than the original. It is also ossible for the new Cantor set to be the same as the original. To see this, reeat the construction in the Remark above, but now make a different choice of gas at each level other than the one with maximal diameter. The only requirement is that at some ste, each ga is eventually selected. On the other hand, if two different sequences yield the same Cantor set, evidently one is a rearrangement of the other. We recall the definitions of Hausdorff measure and dimension. Definition. Let A R be a Borel-measurable set and α > 0. For δ > 0 let } Hδ α A) = inf{ diamei )) α : E i oen, E i A, diame i ) δ. Then, the α-dimensional Hausdorff measure of A, H α A), is defined as H α A) = lim δ 0 H α δ A),
4 C. Cabrelli, U. Molter, V. Paulauskas and R. Shonkwiler and the Hausdorff dimension of A is, dim H A) = su{α : H α A) > 0}. It can be shown [6]), that if in the definition of the Hausdorff measure, the elements of the coverings are chosen to be closed sets, or Borel sets, the resulting measure is the same. If for some choice α = s, 0 < H s A) <, then A is called an s-set. Since we will only be using the Hausdorff dimension in this aer, henceforth we omit the subscrit H. We will mainly consider Cantor sets constructed using the -sequence λ = {λ k } k N, and its rearrangements, where λ k = k. We call such sets -Cantor sets. Sets of similar structure arise in the analysis of the extremal oints of boundaries of random stable zonotoes, which were studied in [2]. However such extremal sets are more comlicated than the Cantor sets treated here since they are random and lie in R d with d 2. In general, the comutation of the Hausdorff dimension or the Hausdorff measure of a set is not easy to do, see [3], [5] and references therein. Estimates from above are usually simler to obtain than estimates from below. In our articular case, showing that the Hausdorff measure is finite, for an aroriate choice of s, will be relatively easy. However, showing that it is ositive will require shar estimates on the size of the intervals of the construction. We now state our main results. Theorem.. Let λ = {λ k } k N be defined by λ k = ), k >. Then Cλ is -set; recisely, a ) 2 H 8 2 ) Cλ ) 2 and furthermore dim C λ =. The second theorem deals with rearrangements of the original sequence λ. Theorem.2. If σλ) is any rearrangement of the sequence λ = k, k N, with >, then 0 dim C σλ). Furthermore, for each 0 < s, there exists a rearrangement σ sλ) such that C σsλ) is an s-set.
Hausdorff Measure of -Cantor Sets 5 We remark here that our main goal is to show that -Cantor sets are s- sets for an aroriate choice of s. If we were only interested in determining the Hausdorff dimension of these sets, we could use the results in the article of Beardon [] about the Hausdorff dimension of what is referred to there as general C-sets; these sets were introduced by Tsuji [7]. The result of Theorem. fits nicely into a result by Falconer [4] g. 55, where he comutes the Box dimension for Cantor sets constructed in this fashion. He shows that the uer and lower Box dimension coincide if and only if the following limit log λ k l = lim k log k exists, in which case the Box dimension is /l. For the articular case of the -series, this limit is, yielding yet another way for obtaining an uer bound for the Hausdorff dimension. 2 Cantor Sets Associated to Geometrical Sequences. In this section, we will briefly leave the -series, and analyze the behavior of Cantor sets associated to sequences with geometric decay. Definition We say that a sequence of ositive terms a = {a k } has at least geometrical decay, if there exist 0 < d < and c > 0 such that a k cd k for all k N. We will show next, that a sequence which tends to zero can be decomosed into finitely many or countably many subsequences, all of them having at least geometrical decay. Lemma 2.. Let a = {a n } be a sequence of ositive terms such that lim a n = 0. Then there exists a family of functions {γ j : N N, j =, 2,... } at most countable such that. γ j is one to one and increasing for all j. 2. γ j N) γ j N) = if j j. 3. N = j γ jn). 4. For all j, the subsequence a j) = {a γjn)} n N has at least geometrical decay.
6 C. Cabrelli, U. Molter, V. Paulauskas and R. Shonkwiler Proof. We define first γ by γ ) = and if γ n) is already defined then γ n + ) = min{m N : m > γ n) and a m /2 n+ }; this being ossible since a n 0. This defines γ inductively. Now we assume that γ,..., γ k are already defined, then if N \ k j= γ jn) is finite, we sto, and redefine γ in such a way that γ N) = N \ k j=2 γ jn). Otherwise, we define γ k+ by γ k+ ) = minn \ k γ j N)) and if γ k+ n) is already defined then j= γ k+ n + ) = min{m N \ k γ j N)) : m > γ k+ n) and a m /2 n+ }. j= If the rocess does not end in a finite number of stes, then N = j γ jn) since every number n must be selected at most at ste n. Let us now rove, that a Cantor set associated to a sequence with at least geometric decay, has Hausdorff dimension 0. Proosition 2.2. Let a = {a k } k N be any sequence such that 0 < a k r k for r <. Then, the Cantor set C a has Hausdorff dimension 0. Proof. We will show, that for each ɛ > 0, dim C a ɛ. Suose that n is sufficiently large and such that j=n+ rj δ. Suose that we removed from I 0 = [0, a k ], n oen intervals of lengths a, a 2,..., a n. What remains can be written as the union of closed intervals E n) j, j =, 2,..., n +. Since n+ j= En) j = j=n+ a j j=n+ rj δ, then {E n) j : j =, 2,..., n + } is a δ-covering of C a. Using the Hölder inequality, we have for each ɛ > 0, ɛ n+ n+ ɛ E n) n + ) ɛ j= E n) j j= j=n+ j r j ɛ r n + ) ɛ n+ ) ɛn = + ) ɛ. r But then n+ lim su E n) j ɛ <, n j=
Hausdorff Measure of -Cantor Sets 7 which roves that dim C a ɛ. Since this is true for every ɛ > 0, we conclude that dim C a = 0. Using this Proosition, together with Lemma 2., we are able to rove the following interesting roerty. Proosition 2.3. Let a = {a n } such that a n > 0 and a n <. Then there exists a rearrangement σa) of a such that dim C σa) = 0. Proof. Using Lemma 2., we can decomose the sequence a into at most countably many subsequences, all of them having at least geometrical decay. Let {γ j } be the family of functions given by Lemma 2. and let C γj be the Cantor set associated to the subsequence {a γjn)}. Note that, since the sequence {a γjn)} has at least geometric decay, dim C γj = 0 by Proosition 2.2. Define now t 0 = 0 and t j = a γjn), j =, 2,.... n Then we have C γj [0, t j ]. Define C to be the union of translates C = ) j C γj + t k ). j The set C is a Cantor set and dim C = 0. Since it is the at most countable union of Cantor sets of dimension zero.) The lengths of the gas of C corresond to the terms of the original sequence a. Then, there is a rearrangement σa) of the sequence, that is associated to the Cantor set C; that is, C = C σa). 3 -Cantor Sets. From this oint on, we will again concentrate on the sequence λ = {k }. First we will show that, for any rearrangement of this sequence, the -Hausdorff measure is finite, immediately roviding an uer bound for the Hausdorff dimension. Proosition 3.. Let C λ be the Cantor set associated to the sequence λ k = k, k N, with >. Then H ) Cλ ) ; in articular, k= dim C λ.
8 C. Cabrelli, U. Molter, V. Paulauskas and R. Shonkwiler Moreover, if σλ) is any rearrangement of this sequence, then dim C σλ). Proof. The roof is analogous to the one of Proosition 2.2. Consider n large enough such that j=n+ λ j δ, so that after removing from I 0 oen intervals of lengths λ, λ 2,..., λ n, we have the closed intervals {E n) j, j =, 2,..., n + }, which is a δ-covering of C. Using the Hölder inequality and the Integral Comarison Test for sequences, we have for 0 < s <, s s n+ n+ s E n) n + ) s = j n + ) s j= E n) j Hence, if s, j= j ) n s n + ) s = j=n+ ) s n + ) s n s s. n+ ) s lim su E n) j s <, n j= and therefore, for any s, Hs δ C λ) <. In articular, dim C λ. To rove the more general case, we can argue as follows. Let σλ) be any rearrangement of λ = {λ k }. Regardless in what order we remove oen intervals from I 0, there will be a ste m at which the first n intervals of length λ, λ 2,..., λ n will be removed m n). We consider the n+ closed intervals which are comlementary in I 0 ) to these removed n-intervals. Clearly E n) j, j =,..., n + again forms a δ-covering of C σλ) and therefore, the same bounds as before hold. Thus Proosition 3. and the uer bounds for Theorems. and.2 are roved. E n) j 3. The Proof of Theorem. Since the bounds from above are roved in Proosition 3., in order to comlete the roof of Theorem., we need to show that the Cantor set C λ has ositive -Hausdorff measure. The following lemma is the main ingredient in the roof of the theorem. We remind the reader that Il k stands for the lth interval obtained in the kth ste of construction, which was described in the Introduction. The length of the interval Il k is given by equation ), from which we make the observation
Hausdorff Measure of -Cantor Sets 9 that, if the gas form a monotone non-increasing sequence, then so do the diameters of these intervals; that is, for l l, I k l Ik l. Lemma 3.2. For all k =, 2,... and l = 0,,..., 2 k, λ 2k +l+ 2 2 2 Ik l 2 2 2 λ 2 +l, k and therefore for l l, Ik l I k l 2. 2) Proof. We can rewrite the exression for Il k in ) as follows Then I k l h=0 =λ 2 k +l I k l = h h=0 j=0 λ 2 k+h +l2 h +j. 2 h 2 h 2 k + l)) = 2 k + l) 2 2 2 = λ 2 k +l 2 2. The bound from below is obtained in a similar way I k l = h h=0 j=0 2 k + l + ) For the inequality 2), since the result follows. λ 2 k+h +l2 h +j h=0 h=0 2 ) Il k Il k Ik 0 I k 2 k, h=0 ) h 2 2 h 2 ) h 2 k + l + 2 h ) h = λ 2k+l+ 2 2 2. The next lemma is a simle algebraic roerty of numbers which we will use reeatedly.
0 C. Cabrelli, U. Molter, V. Paulauskas and R. Shonkwiler Lemma 3.3. Let a, b, c be arbitrary ositive numbers, let > and set x = a + b + c. Then x 2 2 2 c = x a + b. Proof. If x 2 x c 2 c, then 2 2 and, by convexity of the function x, a + b ) x a + b 2. Equivalently, x 2 2, ) a + b, from which the 2 2 result follows. Combining the revious lemmas, we obtain the main relation between the intervals of ste k and those of ste k +. Lemma 3.4. For all k and l = 0,,..., 2 k, I k l I k+ 2l + I k+ 2l+. Proof. By construction and from Lemma 3.2 we have I k l = I k+ 2l + I k+ 2l+ + λ 2 k +l and I k l 2 2 2 λ 2 k +l. Hence, by Lemma 3.3 the result is obtained. Lemma 3.5. Let J be an arbitrary oen interval in I 0. Let k N be fixed. Then 4 J k l. I l : I k l J Proof. If J C λ =, the result is trivial. Now, if J C λ, and J is oen, then there exists some Cantor interval Il k J. Next observe that if Il k and Ik l+ are consecutive intervals from ste k and obtained from one interval in ste k, We shall say that these two intervals have a common father.), then l is even. If we consider two consecutive intervals of ste k not having a common father; i.e., Il k and Ik l+ with l odd, let I be the minimal closed interval containing Il k and Ik l+. Then I Il k ) Ik l+ is a ga of a revious ste; i.e., I = I k l + λ 2s +r + I k l+ l with s k 2 and r = [ ] < l. 2 k s 2 k s We want to rove that I Il k + I k l+. By Lemma 3.2 we have I k l 2 2 2 λ 2 k +l,
Hausdorff Measure of -Cantor Sets and since we conclude that λ 2k +l = 2 k + l) 2 k s ) 2 s + r), I k l 2 2 k s ) 2 2 λ 2 s +r. Since Il+ k < Ik l and k s 2, we have 2 2 ) I 2 k s ) 2 2 + λ 2 s+r 2 2 2 λ 2 s +r. Thus we can aly Lemma 3.3 to obtain I I k l + I k l+. Define now k 0 := min{k N : Il k J for some 0 l < 2k }. First we observe that the interval J can contain at most two intervals of ste k 0. We will rove the case in which J contains exactly two, the other case can be roved similarly. Note that by definition of k 0, J must contain at least one interval of ste k 0.) Let I k0 l and I k0 l+ be the intervals in J. Then l is odd and only four Cantor intervals of ste k 0 can intersect J. These are I k0 l, Ik0 l, Ik0 l+, and Ik0 l+2. Let Ĩ be the smallest interval containing I k0 l and I k0 l+. Using 3.) and since J Ĩ, we have J Ĩ I k 0 l + I k 0 l+. 3) Now using Lemma 3.2 we know that 2 I k0 l Ik0 l ; that is, Finally, since I k0 l+ Ik0 l+2 and Ik0 l+ J, From 3), 4), and 5) we get 2 J 2 I k 0 l I k 0 l. 4) J I k 0 l+ I k 0 l+2. 5) 4 J I k 0 l + I k 0 l + I k 0 l+ + I k 0 l+2. Now, if k k 0, using Lemma 3.4 inductively, we have 4 J I l:i k l J k l
2 C. Cabrelli, U. Molter, V. Paulauskas and R. Shonkwiler and if k < k 0 there are no intervals I k l J, l = 0,..., 2 k and the inequality is obvious. This comletes the roof. We are now ready to rove Theorem.. Let F = {F i } i N be a covering of C λ with oen intervals of length less than δ, Fi C λ and diamf i ) < δ, i. Since C λ is comact, let {F hj = α j, β j )} m j= be a finite subcovering of C λ, F hj F, j =,..., m. Let ε > 0. Since R \ C λ is dense in R, we can construct oen intervals, E j = a j, b j ), j =,..., m such that F hj E j, E j < Fhj + ε m and a j, b j C λ. Therefore, Since ε is arbitrary, if m m E j < F hj + ε. j= j= m E i, 6) 4 2 2) i= we obtain the desired lower bound for the Hausdorff measure. But since a k, b k C λ, for k large enough, we can make Il k so small that, for all 0 l < 2 k, Il k E i for some i. Therefore, using Lemma 3.5, Since Il k Ik l+, we get m m E i Il k 4 i= i= l : Il k Ei 4 2 k Il k 2 k 4 4 Ik 2 k, l=0 and, using the estimate of Lemma 3.2, we get 6). k l=0 I k l.
Hausdorff Measure of -Cantor Sets 3 3.2 The Cantor sets C λ In this section we are going to rove the following Proosition. Proosition 3.6. Let λ = {λ k = k }. For each 0 s rearrangement σ s λ) of λ such that H s C σsλ)) > 0. there exists a To rove this, we first find the Hausdorff dimension of the Cantor set C λ where λ is a articular subsequence of the original -sequence. This result together with the following lemma will comlete the roof. Lemma 3.7. If λ is a subsequence of λ = {λ k = k } such that dim C λ = s, then there exists a rearrangement σλ) = σ s λ) of λ, such that dim C σsλ) = dim C λ = s. Proof. Let γ = {γ k } be the subsequence obtained from λ after deleting the terms of the subsequence λ. By Proosition 2.3 there exists a rearrangement σγ) of γ, such that dim C σγ) = 0. Let now t = k γ k. Then we have C σγ) [0, t ]. Define C = C σγ) C λ + {t }). The set C is a Cantor set and dim C = s since it is the union of one Cantor set of dimension s and one of dimension 0). The lengths of the gas of C corresond to the terms of the original sequence λ. Then, there is a rearrangement σ of the sequence, that is associated to the Cantor set C; that is, C = C σλ). It is now clear that, in order to obtain Proosition 3.6, it suffices for each s to find a articular subsequence λ, such that dim C λ = s. Let x 2 be a fixed real number. We define a subsequence λ m by the following relation: using the fact that m can be decomosed uniquely into m = 2 k + j with k 0 and j = 0,,..., 2 k, and using the notation [x] to denote the greatest integer in x, we set λ m = λ ). 2k +j = λ [xk ]+j = [x k ] + j Since this subsequence is comletely determined by x, and in order to avoid cumbersome notation, we denote by C x the Cantor set C λ.
4 C. Cabrelli, U. Molter, V. Paulauskas and R. Shonkwiler Theorem 3.8. With notation as above and with α, x) := log 2 log x, we have x ) α 4 c H α ) α, x C x ) 2 2 2 for some ositive constant c deending only on x and. Hence, C x is an α-set. Proof. The roof of this theorem has the same flavor as that of Theorem.. However, since we are dealing with subsequences of the original -series, the estimates need some more careful consideration. Again we will slit the roof of the theorem into two searate statements one for the uer bound and one for the lower bound. We use the following notation: all quantities introduced in the roofs of Theorem. and Proosition 3.6 corresonding to the sequence {λ j } will be used with the sign to denote quantities corresonding to { λ j }. Proosition 3.9. Let C x be the Cantor set associated to the sequence λ = { λ 2k +j = λ [xk ]+j, k N, j = 0,,..., 2 k }. Then ) 4 H α α C x ) 2 ; 2 hence dim C x α. Proof. The roof goes along the lines of the roof of Proosition 3., only instead of λ j, we now have λ j. Suose that n = 2 k is sufficiently large such that λ j=n+ j δ. The remaining intervals in Ĩ0 = [0, λ j j ], after removing the oen intervals of lengths λ, λ 2,..., λ n, are contained in the union of intervals of the collection {Ẽn) j : j =, 2,..., n+} which is therefore a δ-covering of C x. Using again the Hölder inequality, we have n+ n+ Ẽn) j s j= Ẽn) j j= We now estimate the quantity j=n+ s λ j = n + ) s = m=k m j=0 j=n+ λ 2 m +j. s λ j n + ) s.
Hausdorff Measure of -Cantor Sets 5 We have m=k m j=0 λ 2m +j = m=k m j=0 λ [x m ]+j = m=k m j=0 [x m ] + j) 2 m [x m ] ) x m 2 m m=k m=k ) m ) k 2 2 2 x = x 2 x x 2 m=k ) k 2 K) x, where K) = 4 /2 2) is a bound for 2x) /x 2). Since 2 k = n + and α = log 2 log x, we have that xk = n + ) /α, and n+ Ẽn) j s K s n + )s ) x ks n + ) s = K s )n + ) s/α, j= which again, if s α, yield, In articular, dim C x α. n+ ) 4 lim su Ẽn) j s s < n 2 < +. 2 j= To comlete the roof that our Cantor set is an α-set, we must show that the Hausdorff measure of C x, H α C x ), is ositive. The idea is to reeat the roofs used for the set C λ. However, it is immediately seen that the same estimate will not work in this case and in fact, this estimate is much harder to obtain. Proosition 3.0. Let >, x > 2 and let λ = { λ k } k N be the sequence defined by λ ). 2 k +j = [x k ]+j If Cx is the Cantor set associated to the sequence λ, and α = log 2 log x, then there exists a ositive constant c, which deends only on x and, such that ) x H α α C x ) c x. 2
6 C. Cabrelli, U. Molter, V. Paulauskas and R. Shonkwiler This Proosition is the analogue of Proosition 3.6, which was roved by combining Lemmas 3.2, 3.3, 3.4 and 3.5. We will need to rovide the analogues of these lemmas for this new sequence. Note that Lemma 3.3 is indeendent of the sequence. Since Ĩ k l = h h=0 j=0 λ 2 k+h +l2 h +j, using the same arguments as in Lemma 3.2 we immediately have the following. Lemma 3.. For every fixed x > 2 and all k and l = 0,..., 2 k, x ) x 2 x λ k+) k l x x 2 x k and therefore, for l l, Ĩk l Ĩk l 2x). Since Lemma 3. rovides an estimate which is not as recise as that of Lemma 3.2, we have now a weaker version of Lemma 3.4. This is robably the essence of the lower bound estimate for the Hausdorff measure. Once we have a relation between the sizes of the intervals of one ste with those of the next ste, we are able to roceed with the bounds for the Hausdorff measure. Lemma 3.2. For all k N and all l = 0,,..., 2 k Ĩk l α B α k Ĩk+ 2l α + Ĩk+ 2l+ α) where the sequence B k satisfies, B k > and ζ, x) > 0. k= B α Proof. From Lemma 3. we get the following estimate Ĩk l x [x k ) ] + l λ2k+l, x 2 x k and since l 2 k, Since Ĩk l = Ĩk+ 2l Ĩk l Ĩk l x x 2 + λ 2k +l + Ĩk+ 2l+ x 2 x x k + 2 k x k x k x k + 2 k ) λ2k+l., this gives the estimate ) ) k = ζ, with ζ = Ĩk+ 2l + Ĩk+ 2l+. 7)
Hausdorff Measure of -Cantor Sets 7 Now ut By simle algebra ) x x k ) ) B k = + 2 x k + 2 k. 2 x B k = x 2 x k ) x x k + 2 k. Since x α = 2, raising both sides of 7) to the ower α, and alying convexity arguments as in Lemma 3.3, we get ) Ĩk l α Ĩk+ 2l α + Ĩk+ 2l+ α 2 Bα k. 2 Therefore Ĩk l α B α k Ĩk+ 2l α + Ĩk+ 2l+ α). Since B k >, to see that k= B α k > 0 it is enough to see that k= B k < +. But x > 2 and > and so we can write x k ) x k + 2 k = ) + 2k + 2 2 x k x )k) + 2 2 x = )k) + 2 2 x )k) + 2 2 x )k) 2 2 x )k + 2 2 x )k) 2 2 x )k. The second inequality follows by an alication of the Mean Value Theorem. Using this, we have that the following series converges ) x 2 k= and therefore k= B k < +. x k ) ) x x k + 2 k 2 2 ) k ) k 2 < +, x Note that the receding roof fails for the case x = 2 because the estimate in Lemma 3. is too imrecise. We now rove the analogue of Lemma 3.5. Lemma 3.3. Let J be an arbitrary oen interval in Ĩ0. Let k N be fixed and again let α = log 2/ log x). Then there exists c, indeendent of k, such that c J α Ĩk l α. l:ĩk l J
8 C. Cabrelli, U. Molter, V. Paulauskas and R. Shonkwiler Proof. If J C λ =, the result is trivial. Otherwise, define k 0 := min{k N : J contains an interval of ste k}. As before, J can contain at most two intervals of ste k 0. Again we will only rove the case in which J contains exactly two. Let Ĩk0 l and Ĩk0 l+ be the intervals in J. Then l is odd and only four Cantor intervals of ste k 0 can intersect J. These are Ĩk0 l, Ĩk0 l, Ĩk0 l+, and Ĩk0 l+2. Since J Ĩk0 l and Ĩk0 l Ĩk0 l+ Ĩk0 l+2, we have 3 J α Ĩk0 l α + Ĩk0 l+ α + Ĩk0 l+2 α Now using Lemma 3., 2x) Ĩk0 l Ĩk0 l ; that is, Hence Since 2x) α J α α 2x) Ĩk0 l α Ĩk0 l α. 3 + 2x) α ) J α Ĩk0 l α + Ĩk0 l α + Ĩk0 l+ α + Ĩk0 l+2 α. Now inductively we aly Lemma 3.2 and get, 3 + 2x) α ) J α B k0 B k0+... B k ) α we have B k0 B k0+... B k ) α k= 3 + 2x) α ζ J α l:ĩk l J l:ĩk l J B α k ζ > 0, Ĩk l α. Ĩk l α. We are now ready to rove Proosition 3.0. Proof of Proosition 3.0 and hence Theorem.2). As in the roof of Theorem., we choose a finite δ-covering consisting of intervals as close as we wish to an arbitrary covering of C x. We must bound the Hausdorff α measure of this covering. Let F = {F i } i N be a covering of C x with oen intervals of length less than δ, Fi C x and diamf i ) < δ, i. Again, let {F hj = α j, β j )} m j= be a finite subcovering of C x, F hj F, j =,..., m and for ε > 0 choose oen intervals, Ẽ j = a j, b j ), j =,..., m such that F hj Ẽj, Ẽj < Fhj + ε m and a j, b j C x.
Hausdorff Measure of -Cantor Sets 9 Therefore, m m Ẽj α < F hj α + ε. j= j= Hence, the roof is comleted by showing that α m i= Ẽi α x c x 2) for some constant c. But again, for k large enough, Ĩk l is so small that, for all l, Ĩl k Ẽi for some i. Therefore using Lemma 3.3, and defining 2c = ζ/3 + 2x) α ), we have m m Ẽi α ζ 3 + 2x) α Ĩk l α 2c i= i= l : Ĩk l Ẽi Since Ĩk l Ĩk l+, we get k l=0 Ĩk l α. k l=0 Ĩk l α 2 k Ĩk 2 k α, and using the estimate of Lemma 3. and the fact that x α = 2, we get m Ẽi α c2 k+ i= 4 A Generalization. x x 2 ) α ) x α x = c k+)α x. 2 In this section we generalize to the case in which r intervals are removed at each ste, r 2; the case r = 2 was considered above. Thus from I 0 r oen intervals are removed leaving the r closed intervals I 0,..., I r. From each of these, r oen intervals are removed and so at end of the second ste of the construction there remains the r 2 closed intervals I 2 0,..., I 2 r 2. Now continue the construction in this fashion. We will denote the associated Cantor set by C r. Surrisingly, the results of the revious sections remain true. Since the roofs can be obtained by the same methodology as resented in the revious sections, we will only state one of the generalized results in this direction. Note that when carrying out the roofs, the role layed by 2 k is now layed by r k since before we had 2 k intervals at a given ste k, and now we have r k intervals at that same ste.
20 C. Cabrelli, U. Molter, V. Paulauskas and R. Shonkwiler Theorem 4.. Let λ = {λ k } k N be defined by λ k = k ), >. Then for r 2, dim C r =. Moreover 0 < H Cr ) < +. References [] A. F. Beardon, On the Hausdorff dimension of general Cantor sets, Proc. Camb. Phil. Soc., 6 965), 679 694. [2] Yu. Davydov, V. Paulauskas, and A. Račkauskas, More on P -Stable Convex Sets in Banach Saces, J. of Theor. Probab., 3, No. 2000), 39 64. [3] K. Falconer, The Geometry of Fractal Sets, Cambridge University Press, 985). [4] K. Falconer, Techniques in Fractal Geometry, John Wiley & Sons, 997). [5] P. Mattila, Geometry of Sets and Measures in Euclidean Saces. Fractals and Rectifiability, Cambridge Studies in Advanced Mathematics, Vol. 44, Cambridge University Press, 995). [6] C. A. Rogers, Hausdorff measures, second ed., Cambridge University Press, Cambridge, UK, 998. [7] M. Tsuji, On the caacity of general Cantor sets, J. Math. Soc. Jaan, 5 953), 235 252.