On the Probabilistic, Metric and Dimensional Theories of the Second Ostrogradsky Expansion

Sergio Albeverio, Iryna Pratsiovyta, Grygoriy Torbin

no. 482

Diese Arbeit ist mit Unterstützung des von der Deutschen Forschungs- gemeinschaft getragenen Sonderforschungsbereichs 611 an der Universität Bonn entstanden und als Manuskript vervielfältigt worden. Bonn, November 2010 ON THE PROBABILISTIC, METRIC AND DIMENSIONAL THEORIES OF THE SECOND OSTROGRADSKY EXPANSION

SERGIO ALBEVERIO1,2,3,4,5, IRYNA PRATSIOVYTA6, GRYGORIY TORBIN7,8

Abstract. We study properties of the random variable η with independent identically distributed differences of digits of the second Ostrogradsky expansion. Necessary and sufficient conditions for η to be discrete resp. singularly continuous are found. We prove that η can not be absolutely continuously distributed. An ergodic theory for the second Ostrogradsky expansions is also developed. In particular, it is proven that for Lebesgue almost all real numbers any digit i from the alphabet A = N appears only finitely many times in the difference-version of the second Ostrogradsky expansion. Properties of a symbolic dynamical system generated by the natural one-sided shift-transformation T on the difference- version of the second Ostrogradsky expansion are also studied. It is shown that there are no probability measures which are invariant and ergodic (w.r.t. T ) and absolutely continuous (w.r.t. Lebesgue measure). We also study properties of subsets belonging to some classes of closed nowhere dense sets defined by characteristic properties of the second Ostrogradsky expansion. In particular, conditions for the 2 set C[O¯ , {Vk}], consisting of real numbers whose Ostrogradsky symbols take values from the sets Vk ⊂ N, to be of zero resp. positive Lebesgue measure are found. Finally, we compare metric, probabilistic and dimensional theories for the second Ostrogradsky expansions with the corresponding theories for the Ostrogradsky-Pierce expansion and for the continued fractions expansion.

1 Institut fürAngewandte Mathematik, UniversitätBonn, Endenicher Allee 60, D-53115 Bonn (Germany); 2HCM and SFB 611, Bonn; 3 BiBoS, Bielefeld–Bonn; 4 CERFIM, Locarno and Acc. Arch., USI (Switzerland); 5 IZKS, Bonn; E-mail: [email protected] 6 National Pedagogical University, Pyrogova str. 9, 01030 Kyiv (Ukraine); E-mail: [email protected] 7 National Pedagogical University, Pyrogova str. 9, 01030 Kyiv (Ukraine) 8 Institute for Mathematics of NASU, Tereshchenkivs’ka str. 3, 01601 Kyiv (Ukraine); E-mail: [email protected] (corresponding author) AMS Subject Classifications (2000): 11K55, 28A80, 37A45, 37B10, 60G30.

1 2 S.ALBEVERIO, I.PRATSIOVYTA, G.TORBIN

Key words: the second Ostrogradsky expansion of real numbers, symbolic dynamical systems, singular probability distributions, Ostrogradsky- Pierce expansion, continued fractions.

1. Introduction The presented paper devoted to the investigation of the expansion of real numbers in the Ostrogradsky series (they were introduced by M. V. Ostrogradsky, a well known ukrainian mathematician who lived from 1801 to 1862). The expansion of x of the form: 1 1 (−1)n−1 x = − + ··· + + ··· , (1) q1 q1q2 q1q2 . . . qn where qn are positive integers and qn+1 > qn for all n, is said to be the expansion of x in the first Ostrogradsky series. The expansion of x of the form: 1 1 (−1)n−1 x = − + ··· + + ··· , (2) q1 q2 qn where qn are positive integers and qn+1 ≥ qn(qn + 1) for all n, is said to be the expansion of x in the second Ostrogradsky series. Each irrational number has a unique expansion of the form (1) or (2). Rational numbers have two finite different representations of the above form (see, e.g., [16]). Shortly before his death, M. Ostrogradsky has proposed two algorithms for the representation of real numbers via alternating series of the form (1) and (2), but he did not publish any papers on this problems. Short Ostrogradsky’s remarks concerning the above representations have been found by E. Ya. Remez [16] in the hand-written fund of the Academy of Sciences of USSR. E. Ya. Remez has pointed out some similarities between the Ostrogradsky series and continued fractions. He also paid a great attention to the applications of the Ostrogradsky series for the numerical methods for solving algebraic equations. In the editorial comments to the book [13] B. Gnedenko has pointed out that there are no fundamental investigations of properties of the above mentioned representations. Since that time there were a lot of publications devoted to the first Ostrogrdsky series (see, e.g., [1, 2] and references and historical notes therein). Unfortunately the metric theory of the second Ostrogradsky is still not well developed. The second Ostrogradsky series converges rather quickly, giving a very good approximation of irrational numbers by rationals, which are partial sums of the above series. The representation of a real number x in the form (2) is said to be 2 2 the O -expansion of x and will be denoted by O (q1(x), . . . , qn(x), ...). ON THE METRIC THEORY OF THE SECOND OSTROGRADSKY EXPANSION3

For a given set of k initial symbols, the (k+1)-th symbol of the O2- expansion can not take the values 1, 2, 3, 4, 5, ..., qk(qk + 1) − 1, which makes these O2-symbols dependent. Let

d1 = q1 and dk+1 = qk+1 − qk(qk + 1) + 1 ∀k ∈ N. Then the previous series can be rewritten in the form ∑ − k+1 ( 1) ¯2 x = =: O (d1(x), d2(x), . . . , dk(x),... ). qk−1(x)(qk−1(x) + 1) − 1 + dk(x) k (3) Expression (5) is said to be the O¯2-expansion (the second Ostrogradsky expansion with independent increments). The number dk = dk(x) is said to be the k-th O¯2-symbol of x. In the O¯2-expansion any symbol (independently of all previous ones) can be chosen from the whole set of positive integers. The main aims of the present paper are: 1) to develop ergodic, metric and dimensional theories of the O¯2 - expansion for real numbers and compare these theories with the corresponding theories for the continued fraction expansion and for the Ostrogradsky-Pierce expansion; 2) to study properties of the symbolic dynamical system generated by the one-sided shift transformation on the O¯2-expansion: ¯2 ∀ x = O (d1(x), d2(x), . . . , dn(x),... ) ∈ [0, 1], ¯2 ¯2 T (x) = T (O (d1(x), d2(x), . . . , dn(x),... )) = O (d2(x), d3(x), . . . , dn(x),... ); 3) to study the distributions of random variables ¯2 η = O (η1, η2, ..., ηn, ...), ¯2 whose O -symbols ηk are independent identically distributed random variables taking the values 1, 2, ... , m, ... with probabilities p1, p2, ... , ∑∞ pm, ... respectively, pm ≥ 0, pm = 1. m=1

2. Properties of cylinders of the O2- and the O¯2-expansions Let us remind that the representation of a real number x in the form ∑ (−1)k−1 x = , where qk(x) ∈ N, qk+1(x) ≥ qk(x)(qk(x) + 1) (4) qk(x) k 2 2 is said to be the O -expansion of x and it is denoted by O (q1(x), . . . , qn(x), ...). Let

d1 = q1 and dk+1 = qk+1 − qk(qk + 1) + 1 ∀k ∈ N. 4 S.ALBEVERIO, I.PRATSIOVYTA, G.TORBIN

Then the previous series can be rewritten in the form ∑ − k+1 ( 1) ¯2 x = =: O (d1(x), d2(x), . . . , dk(x),... ). qk−1(x)(qk−1(x) + 1) − 1 + dk(x) k (5) Expression (5) is said to be the O¯2-expansion (the second Ostrogradsky expansion with independent increments). The number dk = dk(x) is said to be the k-th O¯2-symbol of x. Let (c1, c2, . . . , cn) be a given set of natural numbers. The set O2 { 2 ∈ } ∆c1c2...cn = x : x = O (q1, q2, . . . , qn,...), qi = ci, i = 1, n, qn+j N is said to be the cylinder of rank n with the base c1c2 . . . cn Let us mention some properties of the O2-cylinders. ∪∞ O2 O2 1. ∆c1...cn = ∆c1...cni. i=cn(cn+1) 2m∑−1 − O2 (−1)k 1 1 2 2. inf ∆ − = − = O (c1, . . . , c2m−1, c2m−1(c2m−1+ c1...2m 1 ck c2m−1(c2m−1+1) k=1 1)) = 2 O2 − − ∈ = O (c1, . . . , c2m 2, c2m 1 + 1) ∆c1...c2m−1 ; 2m∑−1 − O2 (−1)k 1 2 O2 sup ∆ − = = O (c1, . . . , c2m−1) ∈ ∆ ; c1...2m 1 ck c1...c2m−1 k=1 ∑2m − O2 (−1)k 1 2 O2 inf ∆ = = O (c1, . . . , c2m) ∈ ∆ ; c1...2m ck c1...c2m k=1 ∑2m − O2 (−1)k 1 1 2 sup ∆ = + = O (c1, . . . , c2m, c2m(c2m+ c1...2m ck c2m(c2m+1) k=1 1)) = 2 O2 − ∈ = O (c1, . . . , c2m 1, c2m + 1) ∆c1...c2m . 3. sup ∆O2 = inf ∆O2 ; c1...c2m−1i c1...c2m−1(i+1) inf ∆O2 = sup ∆O2 . c1...c2mi c1...c2m(i+1)

O2 Lemma 1. The cylinder ∆c ...c is a closed interval [a, b], where { 1 n O (c , . . . , c , c (c + 1)), if n is odd , a = 2 1 n n n O2(c1, . . . , cn), if n is even ; { O (c , . . . , c ), if n is odd , b = 2 1 n O2(c1, . . . , cn(cn + 1)), if n is even .

O2 ⊂ ⊂ O2 Proof. It is clear that ∆c1...cn [a, b]. Let us prove that [a, b] ∆c1...cn . ∈ O2 ∈ Since a, b ∆c1...cn , it is enough to show that any x (a, b)belongs to O2 ∆c1...cn . If a < x < b, then 1 x = a + x1, where a < x1 < b − a = . cn(cn + 1) ON THE METRIC THEORY OF THE SECOND OSTROGRADSKY EXPANSION5

The Ostrogradsky-Remez theorem states that x1 can be decomposed into the second Ostrogradsky series by using the following algorithm:  ≤  1 = q1x + β1 (0 β1 < x),  ≤  q1 = q2β1 + β2 (0 β2 < β1), q2q1 = q3β2 + β3 (0 ≤ β3 < β2),  ......  ≤  qk...q2q1 = qk+1βk + βk+1 (0 βk+1 < βk), ...... i.e.,

1 1 1 1 − − − k−1 x1 = ′ ′ + ′ ... + ( 1) ′ + .... q1 q2 q3 qk ′ ≥ Let us show that q1 cn(cn + 1). Since 1 ≤ 1 − 1 ≤ ≤ 1 ′ ′ ′ x1 ′ , q1 + 1 q1 q2 q1 we get 1 ≤ 1 ′ x1 < , q1 + 1 cn(cn + 1) ′ ≤ ′ and hence, cn(cn + 1) < q1 + 1 and cn(cn + 1) q1. ′ ′ So, the number x has the following O2-expansion: x = O2(c1, . . . , cn, q1, q2,...), ∈ O2 and, therefore, x ∆c1...cn , which proves the lemma. Corollary 1. For the length of the cylinder of rank n the following relations hold:

2 1 | O | → → ∞ ∆c1...... cn = 0 (n ); cn(cn + 1) Corollary 2. The length of a cylinder depends only on the last digit of the base:

2 1 2 |∆O | = = |∆O |, ∀i ≥ max{c (c + 1), s (s + 1)}. c1...cni i(i + 1) s1...smi n n m m Any cylinder of the O2-expansion can be rewritten in terms of the O¯2-expansion: O2 ≡ ¯ O2 ∆c1...cn ∆a1...... an , where a1 = c1, ak = ck + 1 − ck−1(ck−1 + 1), 1 < k < n. Properties of O2-cylinders generate properties of the O¯2-cylinders. Let us mention only some of these properties. Lemma 2. O¯2 | O2 | |∆ | ∆ − 1 1 a1...ani = c1...cn(cn(cn+1) 1+i) ≤ < . O¯2 O2 2 | | |∆ | cn(cn + 1) + i c ∆a1...an c1...cn n 6 S.ALBEVERIO, I.PRATSIOVYTA, G.TORBIN

Proof. O¯2 | O2 | |∆ | ∆ − c (c + 1) a1...ani = c1...cn(cn(cn+1) 1+i) = n n = O¯2 O2 | | |∆ | (cn(cn + 1) + i − 1)(cn(cn + 1) + i) ∆a1...an c1...cn 1 1 1 = ( ) ≤ < . − 2 1 + i 1 (c (c + 1) + i) cn(cn + 1) + i cn cn(cn+1) n n Remark 1. For the continued fractions 1 |∆c.f. | 1 ≤ a1...ani ≤ , 2 | c.f. | 2 3i ∆a1...an i independently of the previous digits.

Lemma 3. For any sequence of positive integers a1, a2, ..., ak the following inequalities hold: ¯2 |∆O | 1 a1a2...akj < , ∀j ∈ N; |∆O¯2 | 22k−1 a1a2...ak ¯2 |∆O | 1 ba2...akj < , ∀b ∈ N, ∀j ∈ N. |∆O¯2 | b2k ba2...ak 2 22 23 Proof. From lemma 2 and from the fact that ck > ck−1 > ck−2 > ck−3 > 2k−2 2k−2 2k−2 . . . > ck−(k−2) > c2 > 2 it follows that ¯2 |∆O | 1 1 1 a1a2...akj < < = . |∆O¯2 | c2 (22k−2 )2 22k−1 a1a2...ak k k−2 ≥ Similarly, taking into account the fact that ck > c2 and c2 b(b + 1) > b2, we get ¯2 |∆O | 1 1 1 1 ba2...akj < < < = , ∀b ∈ N. |∆O¯2 | c2 ck−1 (b2k−1 )2 2k ba2...ak k 2 b

3. Properties of symbolic dynamics related to the O¯2-expansion Let us consider the dynamical system generated by the one-sided shift transformation T on the O¯2-expansion: ¯2 ∀ x = O (d1(x), d2(x), . . . , dn(x),... ) ∈ [0, 1], ¯2 ¯2 T (x) = T (O (d1(x), d2(x), . . . , dn(x),... )) = O (d2(x), d3(x), . . . , dn(x),... ). Recall that a set A is said to be invariant w.r.t. a measurable transformation T, if A = T −1A. A measure µ is said to be ergodic w.r.t. a transformation T, if any invariant set A ∈ B is either of full or of zero ON THE METRIC THEORY OF THE SECOND OSTROGRADSKY EXPANSION7 measure µ. A measure µ is said to be invariant w.r.t. a transformation T, if for any set E ∈ B one has µ(T −1E) = µ(E). It is well known that the development of metric and ergodic theories of some expansion for reals can be essentially simplified if one can find a measure which is invariant and ergodic w.r.t. one-sided shift transformation on the corresponding expansion and absolutely continuous w.r.t. Lebesgue measure. For instance, having the Gauss 1 1 measure (i.e., the measure with the density f(x) = ln 2 1+x on the unit interval) as invariant and ergodic measure w.r.t. the transformation 1 T (x) = x (mod1), one can easily derive main metric and ergodic properties of the continued fraction (c.f.)-expansion. In this section we shall try to find the O¯2-analogue of the Gauss measure. Let Ni(x, k) be the number of the symbols ”i” among the first k symbols of the O¯2-expansion of x. If the limit lim Ni(x,k) exists, then k→∞ k it is said to be the asymptotic frequency of the digit (symbol) "i" in the ¯2 ¯2 O -expansion of the real number x. We shall use the notation νi(x, O ), or νi(x) for cases where no confusion can arise. Theorem 1. In the O¯2- expansion of almost all real numbers from the unit interval any digit i from the alphabet A = N appears only finitely many times, i.e., for Lebesgue almost all real numbers x ∈ [0, 1] and for any symbol i ∈ N one has:

lim sup Ni(x, n) < +∞. n→∞ ¯2 Proof. Let Ni(x) be the number of symbols ”i” in the O -expansion of the real number x. We shall prove that the Lebesgue measure of the set

Ai = {x : Ni(x) = ∞} is equal to zero for any i ∈ N. To this end let us consider the set ¯ k { ¯2 ∈ ∀ ̸ } ∆i = x : x = O (d1, . . . , dk−1, i, dk+1,...); dj N, j = k , of all real numbers whose O¯2-expansion contains the digit i at the k-th position, i.e., dk(x) = i. Lemma 4. For any i ∈ N and k ∈ N one has ( ) 1 1 λ ∆¯ 1 = ≤ , i i(i + 1) 2

1 λ(∆¯ k) ≤ for k > 1. i 22k−2 [ ] ( ) ¯ 1 O¯2 1 1 ¯ 1 1 ≤ 1 Proof. Since ∆i = ∆i = i+1 ; i , we have λ ∆i = i(i+1) 2 . 8 S.ALBEVERIO, I.PRATSIOVYTA, G.TORBIN

From the above mentioned properties of cylindrical sets and from ¯ k the deﬁnition of the set ∆i it follows that ∪∞ ∪∞ ∆¯ k = ... ∆O¯2 i a1...ak−1i a1=1 ak−1=1 and ¯2 |∆O | 1 a1...ak−1i ≤ , |∆O¯2 | 22k−2 a1...ak−1 Therefore   ( ) ∑∞ ∑∞ ∑∞ ∑∞ 2 1 2 1 λ ∆¯ k = ... |∆O¯ | ≤  ... |∆O¯ | = . i a1...ak−1i 22k−2 a1...ak−1 22k−2 a1=1 ak−1=1 a1=1 ak−1=1

{ ¯ k} It is clear that Ai is the upper limit of the sequence of the set ∆i , i.e., ( ) ∩∞ ∪∞ ¯ k ¯ k Ai = lim sup ∆i = ∆i . k→∞ m=1 k=m Taking into account the Borel-Cantelli Lemma and the fact that ∞ ∞ ∑ ∑ 1 λ(∆¯ k) ≤ < +∞, i 22k−2 k=1 k=1 we deduce that

λ(Ai) = 0, ∀i ∈ N. So, ¯ λ(Ai) = 1, ∀i ∈ N. Let ∩∞ ¯ ¯ A = Ai. i=1 Then λ(A¯) = 1, which proves the theorem. Corollary 3. For Lebesgue almost all real numbers x from the unit interval and ∀i ∈ N : νi(x) = 0. −→ Corollary 4. For any stochastic vector p = (p1, p2, . . . , pk,...) the set −→ ¯2 I p = {x : x = O (d1(x), . . . , dk(x),...), νi(x) = pi ∀i ∈ N} is of zero Lebesgue measure. Theorem 2. There are no probability measures which are simultaneously invariant and ergodic w.r.t the one-sided shift transformation T on the O¯2-expansion, and absolutely continuous w.r.t. Lebesgue measure. ON THE METRIC THEORY OF THE SECOND OSTROGRADSKY EXPANSION9

Proof. To prove the theorem ad absurdum, let us assume that there exists an absolutely continuous probability measure ν, which is invariant and ergodic w.r.t. the above deﬁned transformation T . Then for ν-almost all x ∈ [0, 1] (and, so, for a set of positive Lebesgue measure) and for any function φ ∈ L1([0, 1], ν) the following equality holds: ∫ ∫ ∑n−1 1 1 1 j lim φ(T (x)) = φ(x)d(ν(x)) = φ(x)fν(x)dx, n→∞ n j=0 0 0 where fν(x) is the density of the measure ν. ∈ O¯2 Let φi(x) = 1, if x ∆i ; and φi(x) = 0 otherwise. Then the condition ∫ ∫ 1 φi(x)fν(x)dx = fν(x)dx > 0 holds at least for one digit i ∈ N. ∈ O¯2 0 x ∆i

Let the latter condition hold for some ﬁxed number i0. On the other hand we have ∑n−1 1 j Ni0 (x, n) lim φi (T (x)) = lim = 0 n→∞ n 0 n→∞ n j=0 for λ-almost all x ∈ [0, 1]. So, N (x, n) lim i0 = 0 n→∞ n for λ-almost all x ∈ [0, 1], and simultaneously, we have N (x, n) lim i0 > 0 n→∞ n for a set of positive Lebesgue measure. This contradiction proves the theorem. Corollary 5. The application of the classical "king approach" to the development of the metric theory (i.e., the exploitation of the Birkgoﬀ ergodic theorem with a measure which is absolutely continuous w.r.t. Lebesgue measure and which is ergodic and invariant w.r.t. the one- sided shift transformation on the corresponding expansion) is impossible for the case of the O¯2-expansion for real numbers.

4. On singularity of the random O¯2-expansion Let ¯2 η = O (η1, η2, . . . , ηk,... ), (6) be the random O¯2-expansion with independent identically distributed ¯2 O -symbols ηk, i.e. ηk are i.i.d. random variables taking values 1, 2, ... , m, ... with probabilities p1, p2, ... , pm, ... correspondingly, ∑∞ with pm ≥ 0, pm = 1. m=1 10 S.ALBEVERIO, I.PRATSIOVYTA, G.TORBIN

The following theorem completely describes the Lebesgue decompo- sition of the distribution of η. Theorem 3. The random variable η with independent identically distributed increments of the second Ostrogradsy expansion (i.e., the random variable defined by equality (6) is: 1) of pure atomic distribution with a unique atom (iff pi = 1 for some i ∈ N); 2) or it is singularly continuously distributed (in all other cases). Proof. 1) The first statement of the theorem is evident and we added it only for the completeness of the Lebesgue classification. 2) To prove the second statement we firstly remark that the probability measure µη is invariant and ergodic w.r.t. T . So, from the latter Theorem it follows that µη can not be absolutely continuous. Let us show now that in the case of continuity the distribution of η can not contain an absolutely continuous component. Secondly let us remark that a direct application of the strong law of large numbers shows that for µη - almost all x ∈ [0, 1] the following equality holds ¯2 νi(x, O ) = pi, ∀i ∈ N. (7)

Let us choose a positive integer i0 such that pi0 > 0 (there exists at { ∈ least one such a number) and let us consider the set Mi0 = x : x ¯2 } [0, 1], νi0 (x, O ) = pi0 > 0 . From (7) it follows that this set is of full µη-measure. ∗ { ∈ ¯2 } Let us also consider the set Li = x : x [0, 1], νi0 (x, O ) = 0 . 0 ∗ From the corollary after Theorem 1 it follows that λ(Li ) = 1. The ∗ 0 sets Mi0 and Li0 do not intersect. The ﬁrst of them is a support of the probability measure µη, and the second one is the support of the Lebesgue measure on the unit interval. So, the measure µη is purely singular w.r.t. Lebesgue measure, which proves the theorem.

5. Cantor-like sets related to the O¯2-expansion

Let Vk ⊂ N and let ¯2 ¯2 C = C[O , {Vk}] = {x : x = O (d1, d2, . . . , dk,...), dk ∈ Vk, ∀k ∈ N}. The main goal of this section is to study metric and fractal properties ¯2 of the set C[O , {Vk}] and their dependence on the sequence of sets {Vk}. ¯2 It is clear that the set C[O , {Vk}] is nowhere dense if and only if the condition Vk ≠ N holds for inﬁnitely many indices k. Let ¯ F = {x : x ∈ ∆O2 , a ∈ V , j = 1, k}, k a1a2...ak j j ¯ F = {x : x ∈ ∆O2 , a ∈ V , j = 1, k, a ̸∈ V } k+1 a1a2...akak+1 j j k+1 k+1 ON THE METRIC THEORY OF THE SECOND OSTROGRADSKY EXPANSION11

λ(F k+1) = λ(Fk) − λ(Fk+1) λ(F ) λ(F ) k+1 = 1 − k+1 λ(Fk) λ(Fk) Lemma 5. ∞ [ ] ∏ λ(F ) λ(C) = 1 − k . λ(Fk−1) k=1 Proof.

λ(Fk) λ(Fk−1) λ(F2) λ(F1) λ(C) = lim λ(Fk) = lim · · ... · · = →∞ →∞ k k λ(Fk−1) λ(Fk−2) λ(F1) λ(F0) ∞ ∞ [ ] ∏k λ(F ) ∏ λ(F ) ∏ λ(F ) = lim i = k = 1 − k . →∞ k λ(Fi−1) λ(Fk−1) λ(Fk−1) i=1 k=1 k=1 Corollary 6. ( ) ∑∞ ¯2 λ(F k) λ C[O , {Vk}] > 0 ⇔ < +∞; λ(Fk−1) k=1 ( ) ∑∞ ¯2 λ(F k) λ C[O , {Vk}] = 0 ⇔ = ∞. λ(Fk−1) k=1 a) First of all let us consider the case

Vk = {vk + 1, vk + 2,...}, where {vk} is an arbitrary sequence of positive integers.

Theorem 4. Let Vk = {vk + l, l ∈ N}, where vk ∈ N. If there exists a positive integer b such that ∞ ∑ v k+1 < +∞, b2k k=1 ¯2 then λ(C[O , {Vk}]) > 0.

Proof. Let c be an arbitrary positive integer such that c > max{v1, b} O¯2 and let us consider the cylinder ∆c of the ﬁrst rank. Then ( ∩ ) ∑ v∑k+1 O¯2 | | λ F k+1 ∆c = ∆ca2...akj .

a2∈ V2, ..., ak∈ Vk j=1 Taking into account Lemma 3, we have v ∑k+1 1 |∆ | ≤ v · |∆ | < v · · |∆ |, ca2...akj k+1 ca2...ak1 k+1 b2k ca2...ak j=1 12 S.ALBEVERIO, I.PRATSIOVYTA, G.TORBIN and therefore ( ∩ ) ∑ v∑k+1 ¯2 |∆ | λ F ∆O ca2...akj k+1 c a ∈ V , ..., a ∈ V j=1 ( ∩ ) = 2 2 ∑k k ≤ O¯2 | | λ Fk ∆c ∆ca2...ak a2∈ V2, ..., ak∈ Vk ∑ vk+1 | | k ∆ca ...a b2 2 k a ∈ V , ..., a ∈ V v ≤ 2 2 ∑k k = k+1 . | | 2k ∆ca2...ak b a2∈ V2, ..., ak∈ Vk From the construction of the set C[O¯2, {V }] it follows that ∩ k ∩ ¯2 { } ≥ ¯2 { } O¯2 O¯2 λ(C[O , Vk ]) λ(C[O , Vk ] ∆c ) = lim λ(Fk ∆c ) = k→∞ ∩ ∩ ∩ ∩ O¯2 O¯2 O¯2 O¯2 λ(Fk ∩∆c ) ·λ(Fk−1 ∩ ∆c )· ·λ(F2 ∩ ∆c )·λ(F1 ∩ ∆c ) = lim ¯2 ¯2 ... ¯2 ¯2 = k→∞ λ(F − ∆O ) λ(F − ∆O ) λ(F ∆O ) λ(F ∆O ) k 1 c k 2 c [ 1 c ]0 c ∞ ∩ ∞ ∩ ∏ O¯2 ∏ O¯2 λ(Fk ∩∆c ) − λ(F k ∩∆c ) = ¯2 = 1 ¯2 . λ(F − ∆O ) λ(F − ∆O ) k=1 k 1 c k=1 k 1 c Since ( ∩ ) ∞ O¯2 ∞ ∑ λ F k+1 ∆ ∑ ( ∩ c ) ≤ vk+1 λ F ∆O¯2 b2k k=1 k c k=1 ∑∞ ( ) ( ∩ ) vk+1 ¯2 { } ≥ ¯2 { } O¯2 and the series k converges, we have λ C[O , Vk ] λ C[O , Vk ] ∆ > b2 c k=1 0, which proves the Theorem.

Proposition 1. If Vk = {m + 1, m + 2, m + 3, ...} for some m ∈ N, then ¯1 λ(C[O , {Vk}]) > 0; (8)

λ(C[c.f., {Vk}]) = 0; (9) ¯2 λ(C[O , {Vk}]) > 0. (10) Proof. Inequality (8) follows from Theorem 5 of the paper [1]. To prove (9) let us remind that for λ-almost all x ∈ [0, 1] a given digit "i" appears in the continued fraction expansion of the real number 1 · (i+1)2 x with the asymptotic frequency ln 2 ln i(i+2) . This fact is actually a direct corollary from the Birkgoﬀ∫ ergodic theorem and the fact that 1 1 the Gauss measure G(E) := ln 2 1+x dx is invariant and ergodic w.r.t. E one-sided shift transformation on the continued fraction expansion. If i = 1, then 1 4 νc.f.(x) = ln (11) 1 ln 2 3 for λ-almost all x ∈ [0, 1]. On the other hand, c.f. ∀ ∈ { } ν1 (x) = 0, x C[c.f., Vk ], ON THE METRIC THEORY OF THE SECOND OSTROGRADSKY EXPANSION13 which proves equality (9). Finally, inequality (10) is a direct corollary of the latter theorem.

k k k Proposition 2. If Vk = {3 + 1, 3 + 2, 3 + 3, ...}, k ∈ N, then ¯1 λ(C[O , {Vk}]) = 0; (12)

λ(C[c.f., {Vk}]) = 0; (13) ¯2 λ(C[O , {Vk}]) > 0. (14) Proof. Equality (13) follows from the proof of the previous proposition and inequality (14) is a direct consequence of the latter theorem. To prove (12), let us prove two auxiliary lemmas, which have a self standing interest. Lemma 6. Let O¯1 be the Ostrogradskyi-Pierce expansion g1(x)g2(x)...gn(x)... in the diﬀerence form, gn(x) ∈ N. Then for λ-almost all x ∈ [0, 1] one has √ n lim gn(x) ≤ e (15) n→∞ Proof. In [18] it has been proven that for the standard Ostrogradskyi- ¯ Pierce expansion Oq1(x)q2(x)...qn(x)... (qn+1 > qn) for almost all (in the sense of Lebesgue measure) x ∈ [0, 1] one has: √ n lim qn(x) = e, (16) n→∞ where qn(x) = g1(x) + g2(x) + ... + gn(x). From (16) it follows that ∀ε > 0 ∃ N(ε, x) > 0 : ∀n > N(ε, x) n n (e − ε) < qn(x) < (e + ε)

n+1 n+1 (e − ε) < qn+1(x) < (e + ε) n+1 n n+1 So, gn(x) = qn+1(x) − qn(x) < (e + ε) − (e − ε) < (e + ε) . Therefore, √ n 1+ 1 gn(x) < (e + ε) n , ∀n > N(ε, x), and, hence, √ n lim gn(x) ≤ e, n→∞ which proves the lemma. Lemma 7. If

Vk = {vk + l, l ∈ N} (17) and √ k limk→∞ vk > e, (18) then ¯1 λ(C[O , {Vk}]) = 0. 14 S.ALBEVERIO, I.PRATSIOVYTA, G.TORBIN

¯1 Proof. From (17) it follows that gk(x) > vk, ∀x ∈ C[O , {Vk}]. There- fore, √ √ k ≥ k ∀ ∈ ¯1 { } lim gk(x) limk→∞ vk > e, x C[O , Vk ]. k→∞ Taking into account the latter lemma, we get ( ) ¯1 λ C[O , {Vk}] = 0. So, to show inequality (12), it is enough to apply lemma 7 with k vk = 3 . Remark. Proposition 1 shows essential differences between the metric theory of continued fractions and the metric theory of the O¯2- expansion. At the same time this proposition shows some level of similarity between O¯1- and O¯2-expansions of real numbers. Indeed, from (8) and (10) it follows that the deleting of any finite number of digits from the alphabet does not affect the positivity of the Lebesgue ¯1 measure of the set C[O , {Vk}], and therefore, does not change the → Hausdorff∩ dimension of this set. Moreover, both the mapping∩ f1 : E ¯1 ¯2 E C[O , {1, 2, ..., m}] and the mapping f2 : E → E C[O , {1, 2, ..., m}] do not change the Hausdorff dimension of any subset E ⊂ [0, 1]. On the other hand Proposition 2 demonstrates obvious differences in the metric theories of O¯1- and O¯2-expansions. To stress these differences, let us mention that the transformations TO¯2→O¯1 and TO¯2→c.f. are strictly increasing singularly continuous functions on the unit interval, where

O¯2 O¯1 T ¯2→ ¯1 (∆ ) = ∆ , (19) O O g1(x)g2(x)...gk(x)... g1(x)g2(x)...gk(x)... and O¯2 c.f. T ¯2→ (∆ ) = ∆ . (20) O c.f. g1(x)g2(x)...gk(x)... g1(x)g2(x)...gk(x)...

Actually the singularity of the function TO¯2→O¯1 follows from (12) and (14). The singularity of the function TO¯2→c.f. follows from (11) and from the fact that for λ-almost all x ∈ [0, 1] the asymptotic frequency of the digit 1 in the O¯2-expansion of x is equal to zero. So, the continued fraction expansion, O¯1-expansion and O¯2-expansion are "mutually orthogonal", and their metric theories diﬀer essentially. b) Now let us consider the case where

Vk = {1, 2, . . . , mk} (21) and {mk} is an arbitrary sequence of positive integers. Let M1 := 1, and 2 Mk+1 = (Mk + 1) + mk+1, ∀k ∈ N. ∑∞ ( ) M 2 k ¯2 Theorem 5. If < +∞, then λ C[O , {Vk}] > 0. mk+1 k=1 ON THE METRIC THEORY OF THE SECOND OSTROGRADSKY EXPANSION15

Proof. Let ∆O¯2 be an arbitrary cylinder of rank k with a ∈ V , ∀j ∈ a1...ak j j {1, 2, ..., k}. Then ( ∩ ) ∑∞ 2 2 1 λ F ∆O¯ = |∆O¯ | = , k+1 a1...ak a1...aki ck(ck + 1) + mk+1 i=mk+1+1 where ck can be calculated via a1, a2, . . . , ak : ci+1 = ci(ci +1)−1+ai+1 with c1 = a1. It is clear that ( ∩ ) ( ) 2 2 1 λ F ∆O¯ = λ ∆O¯ = , a ∈ V , j = 1, k. k a1...ak a1...ak j j ck(ck + 1) Therefore ( ∩ ) λ F ∆O¯2 1 k+1 a1...ak c (c + 1) ( ∩ ) = ck(ck+1)+mk+1 = k k . O¯2 1 λ Fk ∆ ck(ck + 1) + mk+1 a1...ak ck(ck+1) From (21) it follows that

1 ≤ c1 ≤ m1, 2 ≤ c ≤ c (c + 1) − 1 + m ≤ (m + 1)2 + m , 2 1 1 2 1( 2 ) 2 2 2 2·3 ≤ c3 ≤ c2(c2+1)−1+m3 ≤ (c2+1) +m3 = (m1 + 1) + m2 +m3, ...... 2k−2 2 2 2 2 ≤ ck ≤ (((m1 + 1) + m2) + ... + mk−1) + mk, ... So, 2k−2 2 ≤ ck ≤ Mk, ∀k ∈ N. Therefore ( ∩ ) O¯2 2k−2 2k−2 λ F ∆ 2 (2 + 1) k+1 a1...ak M (M + 1) ≤ ( ∩ ) ≤ k k . 2k−2 2k−2 λ F ∆O¯2 M (M + 1) + m 2 (2 + 1) + mk+1 k a1...ak k k k+1 (22) Since the estimation (22) holds for any cylinder ∆O¯2 , a ∈ V , j = a1...ak j j 1, k, we get − 22k 1 λ(F ) (M + 1)2 ≤ k+1 ≤ k . 2k−1 2 2 + mk+1 λ(Fk) (Mk + 1) + mk+1 ∞ ∞ ∑ 2 ∑ (Mk+1) ∞ λ(F k+1) ∞ If 2 < + , then < + , and taking into (Mk+1) +mk+1 λ(Fk) k=1 k=1 ( ) ¯2 account corollary 6 we deduce that λ C[O ,Vk] > 0. One can easily verify that ( ) 2 M +1 2 2 2 k · 1 M (M + 1) M Mk = k ≤ k = ( ) k ≤ mk+1 2 2 2 1 + M 2 Mk + mk+1 (Mk + 1) + mk+1 Mk+1 2 k · M + mk+1 Mk k 4M 2 1 4M 2 ≤ k ≤ ≤ k 2 mk+1 . 4Mk + mk+1 1 + 2 mk+1 4Mk 16 S.ALBEVERIO, I.PRATSIOVYTA, G.TORBIN

So, ∞ ∞ ∑ (M + 1)2 ∑ M 2 k ∞ ⇔ k ∞ 2 < + < + . (Mk + 1) + mk+1 mk+1 k=1 k=1 ∑∞ M 2 Therefore the convergence of the series k implies the positivity mk+1 k=1 ¯2 of the Lebesgue measure of the set C[O , {Vk}].

∑∞ − ( ) 22k 1 2 Theorem 6. If − = +∞, then λ C[O¯ , {V }] = 0. 22k 1+m k k=1 k+1 Proof. Using the estimation (22) we have ( ∩ ) O¯2 − λ F k+1 ∆ 2k 1 ( ∩ a1...a)k ≥ 2 λ F ∆O¯2 2k−1 k a1...ak 2 + mk+1 for all cylinders ∆O¯2 , a ∈ V , j = 1, k. a1...ak j j Therefore ( ) − λ F 22k 1 k+1 ≥ , ∀k ∈ N. 2k−1 λ(Fk) 2 + mk+1 ∑∞ − ( ) 22k 1 2 If − = +∞, then, applying corollary 6, we get λ C[O¯ , {V }] = 22k 1+m k k=1 k+1 0, which proves the Theorem

2k−1 Proposition 3. Let Vk = {1, 2, ..., 2 }. Then ¯1 λ(C[O , {Vk}]) > 0; (23)

λ(C[c.f., {Vk}]) > 0; (24) ¯2 λ(C[O , {Vk}]) = 0. (25) Proof. Equality (25) is a direct corollary of the latter theorem. To prove inequality (23), let us remind (see, e.g., [5]) that the con- ∑∞ dition m1+m2+...+mk < +∞ implies the positivity of the Lebesgue mk+1 k=1 ¯1 measure of the set C[O , {Vk}] with Vk = {1, 2, ..., mk}. Since the series ∞ ∑ k−1 m1+m2+...+mk 2 diverges for mk = 2 , we get (23). mk+1 k=1 To prove inequality (24), let us remind (see, e.g., [13]) that 1 |∆c.f. | 1 ≤ a1...ani ≤ . 2 | c.f. | 2 3i ∆a1...an i So, ∑ |∆c.f. | a1...ani ∑∞ i̸∈V 2 4 k+1 ≤ < , c.f. 2 2k−2 |∆a ...a | − i 2 1 n i=22k 1 +1 ON THE METRIC THEORY OF THE SECOND OSTROGRADSKY EXPANSION17 and therefore ∞ c.f. ∞ ∑ λ(F ) ∑ 4 k+1 ≤ < +∞, c.f. 22k−2 k=1 λ(Fk ) k=1 which implies the positivity of the Lebesgue measure of the set λ(C[c.f., {Vk}]).

c) Finally let us consider the case where both the set Vk and the set V k := N \ Vk are inﬁnite for any kinN. \{ (k) (k) (k) } { (k)}∞ Theorem 7. Let Vk = N b1 , b2 , . . . , bm ,... , where bm m=1 is an increasing sequence of positive integers ∀k ∈ N, and let for any k ∈ N exist a positive integer dk ∈ N such that k − k ≤ ∀ ∈ bn+1 bn dk, n N. (26) Then if ∞ ∑ 1 = +∞, (27) b(k) · d ( ) k=1 1 k ¯2 then λ C[O , {Vk}] = 0. ¯ Proof. Let ∆O2 be an arbitrary cylinder of rank k − 1 with a ∈ a1a2...ak−1 j Vj, ∀j ∈ {1, 2, . . . , k − 1}. Then | O¯2 | ≥ | O¯2 | ≥ ≥ | O¯2 | ∆ (k) ∆ (k) ... ∆ (k)− . a1a2...ak−1b1 a1a2...ak−1(b1 +1) a1a2...ak−1(b2 1) Therefore (k)− b2∑1 2 1 2 |∆O¯ | ≥ · |∆O¯ | (k) (k) (k) a1a2...ak−1i a1a2...ak−1b1 − − b2 b1 1 (k) i=b1 +1 On the other hand,

(k)− (k)− b1∑1 b1∑1 2 1 | O¯ | ∆a a ...a − i = = 1 2 k 1 (c − (c − + 1) − 1 + i)(c − (c − + 1) + i) i=1 i=1 k 1 k 1 k 1 k 1

b(k)−1 ( ) 1∑ 1 1 = − = c − (c − + 1) − 1 + i c − (c − + 1) + i i=1 k 1 k 1 k 1 k 1 1 1 b(k) − 1 = − = 1 . c − (c − + 1) (k) − (k) − k 1 k 1 ck−1(ck−1 + 1) + b1 1 ck−1(ck−1 + 1)(ck−1(ck−1 + 1) + b1 1) Hence (k)− b1∑ 1 (k) 2 b −1 |∆O¯ | ( 1 ) a1a2...ak−1i (k) ck−1(ck−1+1) ck−1(ck−1+1)+b −1 i=1 = 1 = | O¯2 | ( 1)( ) ∆ (k) (k)− (k) a1a2...ak−1(b1 ) ck−1(ck−1+1)+b1 1 ck−1(ck−1+1)+b1 18 S.ALBEVERIO, I.PRATSIOVYTA, G.TORBIN ( ) (k) ( ) (k) ck−1(ck−1 + 1) + b1 · (k) − b1 (k) − ≤ = b1 1 = 1 + (b1 1) ck−1(ck−1 + 1) ck−1(ck−1 + 1) ( ) (k) b1 (k) ≤ 1 + · b =: lk. 22k−2 1 So,  b(k)−1  1∑  | O¯2 | ≥ 1 · | O¯2 |  ∆ (k) ∆a a ...a − i ; a a ...a − b lk 1 2 k 1 1 2 k 1 1 i=1 (k)− (28)  b2∑ 1  2 2  |∆O¯ | ≥ 1 · |∆O¯ |.  (k) (k)− (k) a1a2...ak−1i a1a2...ak−1b1 b2 b1 (k) i=b1 +1

(k)− b3∑1 2 1 2 |∆O¯ | ≥ · |∆O¯ |; (k) (k) (k) a1a2...ak−1i a1a2...ak−1b2 − b3 b2 (k) i=b2 +1 ......

b(k) −1 b(k) −1 n∑+1 n∑+1 O¯2 1 O¯2 1 O¯2 |∆ (k) | ≥ · |∆ | ≥ · |∆ |. a a ...a − b (k) (k) a1a2...ak−1i a1a2...ak−1i 1 2 k 1 n − dk bn+1 bn (k) (k) i=bn +1 i=bn +1 From (28) it follows that

(k)− (k)− b1∑1 b2∑1 O¯2 1 O¯2 1 O¯2 |∆ (k) | ≥ · |∆ | + |∆ | ≥ a a ...a − b a1a2...ak−1i a1a2...ak−1i 1 2 k 1 1 2lk 2dk i=1 (k) i=b1 +1

(k)− b2∑1 1 2 ≥ |∆O¯ |. a1a2...ak−1i 2lk · dk ̸ (k) i=1, i=b1 Therefore, ∑ ∑ 2 1 2 |∆O¯ | ≥ |∆O¯ |, a1a2...ak−1i a1a2...ak−1i 2lkdk i̸∈Vk i∈Vk

∀(a1, a2, . . . , ak−1), aj ∈ Vj, j ∈ {1, 2, . . . , k − 1}. Hence { 1 λ(F k) ≥ · λ(Fk); 2lkdk λ(F k) + λ(Fk) = λ(Fk−1). So, λ(F ) 1 1 k ≥ ≥ . λ(Fk−1) 2lkdk + 1 4lkdk ON THE METRIC THEORY OF THE SECOND OSTROGRADSKY EXPANSION19 ∑∞ ∑∞ 1 λ(F k) ¯2 If = +∞, then = +∞, and, therefore, λ(C[O , {Vk}]) = lk·dk λ(Fk−1) k=1 k=1 0. It is clear that ∞ ∑ 1 ∑ 1 ∑ 1 = ( ) + ( ) , l d (k) (k) k k ∈ b1 (k) ̸∈ b1 (k) k=1 k A 1 + − · b · d k A 1 + − · b · d 22k 2 1 k 22k 2 1 k { } (k) b1 where A = k : − ≤ 1 . 22k 2 ∑ ∑∞ ( 1 ) 1 The series (k) diverges if and only if the series (k) b b ·d k∈A 1 · (k)· k=1 1 k 1+ k−2 b1 dk 22 does, and the series ∑ ( 1) (k) ̸∈ b1 (k) k A 1 + − · b · d 22k 2 1 k always converges. ∑∞ Therefore the divergence of the series 1 is equivalent to the lk·dk k=1 ∑∞ divergence of the series 1 , which proves the Theorem. b(k)·d k=1 1 k

Corollary 7. Let Vk = N \{b1, b2, . . . , bm,...}. If ∃ d ∈ N : b − b ≤ d ∀n ∈ N, ( ) n+1 n ¯2 then λ C[O , {Vk}] = 0. ( ) ¯2 Corollary 8. If Vk = N \{1, 3, 5,...}, then λ C[O , {Vk}] = 0. ( ) 2 ¯2 Theorem 8. If Vk = N \{1, 4, 9, . . . , m ,...}, then λ C[O , {Vk}] > 0. ¯ Proof. Let ∆O2 be an arbitrary cylinder of rank k − 1 with a ∈ a1a2...ak−1 j Vj, ∀j ∈ {1, 2, . . . , k − 1}. Then ( ∩ ) ∑∞ O¯2 1 λ F k ∆a ...a − = , 1 k 1 (c − (c − + 1) − 1 + m2)(c − (c − + 1) + m2) m=1 k 1 k 1 k 1 k 1

2 1 |∆O¯ | = ,. a1...ak−1 ck−1(ck−1 + 1) ( ∩ ) λ F ∆O¯2 ∑∞ k a1...ak−1 c − (c − + 1) ( ) = k 1 k 1 < 2 − 2 2 λ ∆O¯ (ck−1(ck−1 + 1) 1 + m )(ck−1(ck−1 + 1) + m ) a1...ak−1 m=1 ∞ ∞ ∑ 2 ∑ c − (c − + 1) − 1 + m 1 < k 1 k 1 = < (c − (c − + 1) − 1 + m2)(c − (c − + 1) + m2) c − (c − + 1) + m2 m=1 k 1 k 1 k 1 k 1 m=1 k 1 k 1 20 S.ALBEVERIO, I.PRATSIOVYTA, G.TORBIN

∫+∞ 1 π π < dx = √ ≤ 2 · 2k−2 ck−1(ck−1 + 1) + x 2 · (ck−1(ck−1 + 1)) 2 2 0 Therefore λ(F ) π k < . 2k−2 λ(Fk−1) 2 · 2 Since ∞ ∑ λ(F ) k < +∞, λ(Fk−1) ( ) k=1 ¯2 we get λ C[O ,Vk] > 0,which proves the theorem.

6. Fractal properties of real numbers with bounded O¯2-digits In the case of zero Lebesgue measure, the next level for the study of ¯2 properties of the sets C[O , {Vk}] is the determination of their Haus- dorﬀ dimension dimH (·) (see, e.g., [8] for the deﬁnition and main properties of this main fractal dimension). We shall study this problem for the case where Vk = {1, 2, ..., mk}. A similar problem for the continued fraction expansion were studied by many authors during last 60 years. Set c.f. E2 = {x : x = ∆ , αk(x) ∈ {1, 2}}. α1(x)...αk(x)... In 1941 Good [9] shows that

0, 5194 < dimH (E2) < 0, 5433. In 1982 and 1985 Bumby [6, 7] improves these bounds:

0, 5312 < dimH (E2) < 0, 5314. In 1989 Hensley [10] shows that

0, 53128049 < dimH (E2) < 0, 53128051. In 1996 the same author ([11]) improves his estimate up to 0, 5312805062772051416. A new approach to the determination of the Hausdorﬀ dimension of the set E2 with a desired precision was developed by Jenkinson and Policott in 2001 [12]. Our nearest aim is to study fractal properties of sets which are O¯2- ¯2 analogues of the above discussed set E2, i.e., the set C[O , {1, 2}] and its generalization C[O¯2, {1, 2, ..., m}]. The following theorem shows that ¯2 from the fractal geometry point of view the sets E2 and C[O , {1, 2}] (as well as their generalizations) are cardinally diﬀerent. ON THE METRIC THEORY OF THE SECOND OSTROGRADSKY EXPANSION21

Theorem 9. Let Vk = {1, 2, 3, ..., mk}, where mk ∈ N. If for any positive α the following equality m1 · m2 · ... · mk lim k−1 = 0 (29) k→∞ 2(α2 ) ¯2 holds, then the Hausdorﬀ dimension of the set C[O , {Vk}] is equal to zero, i.e., ¯2 dimH (C[O , {Vk}]) = 0.

Proof. From the construction of the sets Vk it follows that the set ¯2 C[O , {Vk}] can be covered by m1 cylinders of the ﬁrst rank, by m1 ·m2 cylinders of rank 2, ..., by m1 · m2 · ... · mk cylinders of rank k, ... . The cylinder ∆|11{z... 1} has the maximal length among all cylinders of k rank k: 1 1 |∆ | ≤ < . 11 ... 1 (2k−1) | {z } ck(ck + 1) 2 k ¯2 Let us consider the coverings of the set C[O , {Vk}] by cylinders of rank k: ∆ , a ∈ V , a ∈ V , . . . , a ∈ V . (a1a2...ak 1 ) 1 2 2 k k ∪ ¯2 It is clear that ⊃ C[O , {Vk}]. a1∈V1,...,ak∈Vk 1 For a given ε > 0 there exists k ∈ N such that − < ε. In such 2(2k 1) a case the length of any k-rank cylinder of the O¯2-expansion does not exceed ε. So, ∑ ∑ α ¯2 { } | |α ≤ | |α ≤ Hε (C[O , Vk ]) = inf∪ Ei ∆a1a2...ak |Ei|≤ε,( Ei)⊃C i a ∈V ,...,a ∈V ( ) 1 1 k k 1 α m · m · ... · m ≤ m · m · ... · m · = 1 2 k . 1 2 k 2(2k−1) 2α(2k−1) α ¯2 { } α ¯2 { } α ¯2 { } H (C[O , Vk ]) = lim Hε (C[O , Vk ]) = lim Hε (C[O , Vk ]), ε↓0 k→∞ k where 1 ε = . k 2(2k−1) Since m · m · ... · m α ¯2 { } ≤ 1 2 k H (C[O , Vk ]) − , εk 2α(2k 1) we have m · m · ... · m α ¯2 { } α ¯2 { } ≤ 1 2 k ∀ H (C[O , Vk ]) = lim Hε (C[O , Vk ]) lim k−1 = 0 ( α > 0) k→∞ k k→∞ 2α(2 ) . α ¯2 Therefore, H (C[O , {Vk}]) = 0, ∀α > 0, and so ¯2 α ¯2 dimH (C[O , {Vk}]) = inf{α : H (C[O , {Vk}]) = 0} = 0. 22 S.ALBEVERIO, I.PRATSIOVYTA, G.TORBIN

k Corollary 9. If there exists a number a ∈ N such that mk ≤ a , ∀k ∈ N, then ¯2 dimH (C[O , {Vk}]) = 0.

Corollary 10. If mk = m0, ∀k ∈ N for some positive integer m0, then ¯2 dimH (C[O , {Vk}]) = 0. Let B(O¯2) be the set of all real numbers with bounded O¯2-symbols, i.e., ¯2 { ∃ ∈ ≤ ∀ ∈ } B(O ) = x : x = ∆a1(x)...ak(x)... : K(x) N : aj(x) K(x), j N . Theorem 10. The set B(O¯2) of all numbers with bounded O¯2-symbols is an anomalously fractal set, i.e., the Hausdorﬀ dimension of B(O¯2) is equal to 0: ¯2 dimH B(O ) = 0. Proof. The set B(O¯2) can be decomposed in the following way: ∪∞ ¯2 ¯2 B(O ) = Bi(O ), i=1 where ¯2 Bi(O ) = {x : aj(x) ≤ i, ∀j ∈ N}. Since, ¯2 dimH Bi(O ) = 0, ∀i ∈ N, we have ¯2 ¯2 dimH B(O ) = sup dimH Bi(O ) = 0, i which proves the theorem. Remark 2. From [14] it follows that the set of continued fractions with bounded partial quotients is of full Hausdorﬀ dimension:

dimH (B(c.f.)) = 1, which stresses the essential diﬀerence between the dimensional theories for the O¯2-expansion and the continued fraction expansion. Acknowledgement This work was partly supported by DFG 436 UKR 113/97 project and by the Alexander von Humboldt Foundation.

References

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