Expansions of numbers in positive Lüroth series and their applications to metric, probabilistic and fractal theories of numbers

Expansions of numbers in positive Lüroth series and their applications to metric, probabilistic and fractal theories of numbers

We describe the geometry of representation of numbers belonging to (0, 1] by the positive Lüroth series, i.e., special series whose terms are reciprocal of positive integers. We establish the geometrical meaning of digits, give properties of cylinders, semicylinders and tail sets, metric relations;...

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Hauptverfasser: Zhykharyeva, Yu., Pratsiovytyi, M.
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Zitieren:Expansions of numbers in positive Lüroth series and their applications to metric, probabilistic and fractal theories of numbers / Yu. Zhykharyeva, M. Pratsiovytyi // Algebra and Discrete Mathematics. — 2012. — Vol. 14, № 1. — С. 145–160. — Бібліогр.: 18 назв. — англ.

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spelling irk-123456789-1522352019-06-10T01:26:15Z Expansions of numbers in positive Lüroth series and their applications to metric, probabilistic and fractal theories of numbers Zhykharyeva, Yu. Pratsiovytyi, M. We describe the geometry of representation of numbers belonging to (0, 1] by the positive Lüroth series, i.e., special series whose terms are reciprocal of positive integers. We establish the geometrical meaning of digits, give properties of cylinders, semicylinders and tail sets, metric relations; prove topological, metric and fractal properties of sets of numbers with restrictions on use of “digits”; show that for determination of Hausdorff-Besicovitch dimension of Borel set it is enough to use connected unions of cylindrical sets of the same rank. Some applications of L-representation to probabilistic theory of numbers are also considered. 2012 Article Expansions of numbers in positive Lüroth series and their applications to metric, probabilistic and fractal theories of numbers / Yu. Zhykharyeva, M. Pratsiovytyi // Algebra and Discrete Mathematics. — 2012. — Vol. 14, № 1. — С. 145–160. — Бібліогр.: 18 назв. — англ. 1726-3255 2010 MSC:11K55. http://dspace.nbuv.gov.ua/handle/123456789/152235 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
description We describe the geometry of representation of numbers belonging to (0, 1] by the positive Lüroth series, i.e., special series whose terms are reciprocal of positive integers. We establish the geometrical meaning of digits, give properties of cylinders, semicylinders and tail sets, metric relations; prove topological, metric and fractal properties of sets of numbers with restrictions on use of “digits”; show that for determination of Hausdorff-Besicovitch dimension of Borel set it is enough to use connected unions of cylindrical sets of the same rank. Some applications of L-representation to probabilistic theory of numbers are also considered.
format Article
author Zhykharyeva, Yu.
Pratsiovytyi, M.
spellingShingle Zhykharyeva, Yu.
Pratsiovytyi, M.
Expansions of numbers in positive Lüroth series and their applications to metric, probabilistic and fractal theories of numbers
Algebra and Discrete Mathematics
author_facet Zhykharyeva, Yu.
Pratsiovytyi, M.
author_sort Zhykharyeva, Yu.
title Expansions of numbers in positive Lüroth series and their applications to metric, probabilistic and fractal theories of numbers
title_short Expansions of numbers in positive Lüroth series and their applications to metric, probabilistic and fractal theories of numbers
title_full Expansions of numbers in positive Lüroth series and their applications to metric, probabilistic and fractal theories of numbers
title_fullStr Expansions of numbers in positive Lüroth series and their applications to metric, probabilistic and fractal theories of numbers
title_full_unstemmed Expansions of numbers in positive Lüroth series and their applications to metric, probabilistic and fractal theories of numbers
title_sort expansions of numbers in positive lüroth series and their applications to metric, probabilistic and fractal theories of numbers
publisher Інститут прикладної математики і механіки НАН України
publishDate 2012
url http://dspace.nbuv.gov.ua/handle/123456789/152235
citation_txt Expansions of numbers in positive Lüroth series and their applications to metric, probabilistic and fractal theories of numbers / Yu. Zhykharyeva, M. Pratsiovytyi // Algebra and Discrete Mathematics. — 2012. — Vol. 14, № 1. — С. 145–160. — Бібліогр.: 18 назв. — англ.
series Algebra and Discrete Mathematics
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first_indexed 2025-07-13T02:37:01Z
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fulltext Jo ur na l A lg eb ra D is cr et e M at h.Algebra and Discrete Mathematics RESEARCH ARTICLE Volume 14 (2012). Number 1. pp. 145 – 160 c© Journal “Algebra and Discrete Mathematics” Expansions of numbers in positive Lüroth series and their applications to metric, probabilistic and fractal theories of numbers Yulia Zhykharyeva and Mykola Pratsiovytyi Communicated by A. P. Petravchuk Abstract. We describe the geometry of representation of numbers belonging to (0, 1] by the positive Lüroth series, i.e., special series whose terms are reciprocal of positive integers. We establish the geometrical meaning of digits, give properties of cylinders, semi- cylinders and tail sets, metric relations; prove topological, metric and fractal properties of sets of numbers with restrictions on use of “digits”; show that for determination of Hausdorff-Besicovitch dimension of Borel set it is enough to use connected unions of cylin- drical sets of the same rank. Some applications of L-representation to probabilistic theory of numbers are also considered. Introduction There exist many models of real number based on positive integers. One of them is a model of number in the form of (finite and infinite) regular continued fraction. Today they study and use different models of number in the form of convergent series (number is a series, number is a sum of series, number is expanded in series). Mostly of these series are positive or alternating. Engel [12], Sylvester [16], Lüroth [8, 3, 4, 5, 7, 13], Ostrogradsky [9, 1, 2], Sierpiński [15], Pierce series et al. are among them. 2010 MSC: 11K55. Key words and phrases: Lüroth series, L-representation, cylinder, semicylin- der, shift operator, random variable defined by L-representation, fractal, Hausdorff- Besicovitch dimension. Jo ur na l A lg eb ra D is cr et e M at h. 146 Expansions of numbers in positive Lüroth series Some of them have relatively simple self-similar geometry [13, 17, 18, 14], but other have rather complicated and non-self-similar geometry [1, 2, 9, 12, 10]. Such expansions of numbers can be represented in different forms using positive integers. It is an encoding of number with symbols of infinite alphabet. Lüroth [8] introduced in 1883 expansion of x ∈ (0, 1] in special positive series such that its terms are reciprocal to positive integers. Geometry of this expansion of numbers is self-similar and convenient for modelling of mathematical objects with non-trivial topological, metric and fractal local properties based on relatively simple metric relations generated by cylindrical sets. In papers [17, 18] we particularly studied properties of cylindrical sets and used them for study of one class of infinite Bernoulli convolutions. In this paper we continue to study geometry of this expansion. In particular, we study properties of semicylinders, supercylinders and tail sets, solve some problems of metric and fractal theories of numbers, provide some applications of results. 1. L-representation of real numbers Theorem 1. Any number x ∈ (0, 1] can be uniquely expanded in Lüroth series, i.e., for x exists a unique sequence of positive integers dn = dn(x) such that x = 1 d1 + 1 + ∞ ∑ n=2 1 Dn−1(dn + 1) ≡ ∆L d1d2...dn... , (1) where Dn ≡ d1(d1 + 1)d2(d2 + 1) . . . dn(dn + 1). Proof. Existence. Since (0; 1] = ∞ ⋃ n=1 ( 1 n+1 , 1 n ], it is evident that there exists d1 such that 1 d1+1 < x 6 1 d1 . Then 0 < x − 1 d1 + 1 ≡ x1 ≤ 1 d1 − 1 d1 + 1 = 1 d1(d1 + 1) = 1 D1 . Since (0; 1 D1 ] = ∞ ⋃ n=1 ( 1 D1(n+1) , 1 D1n ], it is evident that for x1 ∈ (0; 1 d1(d1+1) ] there exists d2 ∈ N such that 1 d1(d1+1)(d2+1) < x1 ≤ 1 d1(d1+1)d2 . Whence it follows that 0 < x1 − 1 d1(d1 + 1)(d2 + 1) ≡ x2 ≤ 1 d1(d1 + 1)d2(d2 + 1) = 1 D2 Jo ur na l A lg eb ra D is cr et e M at h. Yu. Zhykharyeva, M. Pratsiovytyi 147 and x = 1 d1 + 1 + x1 = 1 d1 + 1 + 1 d1(d1 + 1)(d2 + 1) + x2. Let us perform analogous arguments for x2 and so on to infinity and obtain (1). Series (1) is convergent because of xm = 1 d1(d1 + 1)d2(d2 + 1) . . . dm(dm + 1) = 1 Dm < 1 2m → 0 (m → ∞). Uniqueness. Suppose that x has at least two different expansions in the form (1): x = ∆L d1...dm−1dmdm+1... = ∆L d1...dm−1d′ md ′ m+1 ... , dm 6= d′ m. Without loss of generality we assume that d′ m < dm. Then δ ≡ ∆L d1...dm−1d′ md ′ m+1 ... − ∆L d1...dm−1dmdm+1... = 1 Dm · δ1, δ1 ≡ 1 dm + 1 − 1 d′ m + 1 + ∞ ∑ n=1 1 D′ m+n−1(d′ m+n + 1) − ∞ ∑ n=1 1 Dm+n−1(dm+n + 1) . However, δ1 > ( dm − d′ m (d′ m + 1)(dm + 1) − ∞ ∑ n=1 1 Dm+n−1(dm+n + 1) ) ≥ ≥ 1 (d′ m + 1)(dm + 1) − 1 dm(dm + 1) ( 1 2 + 1 22 + 1 23 + . . . ) = 0. Thus, δ1 > 0. This contradicts the assumption that there are two different expansions of the same number. Definition 1. Brief notation x = ∆L d1d2...dn... of the expansion (1) is called L-representation of x, and dn = dn(x) is its nth L-symbol. Theorem 2 ([17]). A number x ∈ (0, 1] is rational if its L-representation is periodic. 2. Geometry of L-representation: cylinders and semicylinders Definition 2. Let (c1, c2, . . . , cm) be a fixed m-tuple of positive integers. Cylinder of rank m with the base c1c2 . . . cm is a set ∆L c1c2...cm := {x : x = ∆L c1c2...cmdm+1dm+2... , dn+i ∈ N}. Jo ur na l A lg eb ra D is cr et e M at h. 148 Expansions of numbers in positive Lüroth series Cylinders have the following properties. 1. ∆L c1c2...cm = ∞ ⋃ i1=1 . . . ∞ ⋃ ik=1 ∆L c1...cmi1i2...ik ∀k ∈ N . 2. Cylinder ∆L c1c2...cm is a half-interval with endpoints inf ∆L c1...cm = 1 c1 + 1 + 1 b1(c2 + 1) + . . . + 1 bm−1(cm + 1) = am; max ∆L c1...cm = am + 1 bm , where bm = c1(c1 + 1) . . . cm(cm + 1). 3. The length of cylinder is equal to ∣ ∣ ∣∆L c1...cm ∣ ∣ ∣ = 1 c1(c1 + 1) . . . cm(cm + 1) = m ∏ i=1 1 ci(ci + 1) . 4. For any sequence of positive integers (cn), the intersection ∞ ⋂ m=1 ∆L c1c2...cm... = x ≡ ∆L c1c2...cm... ∈ (0, 1]. 5. If dj(a) = dj(b) for j < m and dm(a) > dm(b), then a < b. 6. Rearrangement of L-symbols in the base does not change the length of cylinder. 7. Basic metric relation: ∣ ∣ ∣∆L c1...cm ∣ ∣ ∣ = i(i + 1) ∣ ∣ ∣∆L c1...cmi ∣ ∣ ∣. 8. ∞ ∑ j=a ∣ ∣ ∣∆L c1...cmj ∣ ∣ ∣ = a ∣ ∣ ∣∆L c1...cm ∣ ∣ ∣. 9. ∣ ∣ ∣∆L c1...cma ∣ ∣ ∣ = ∞ ∑ j=a(a+1) ∣ ∣ ∣∆L c1...cmj ∣ ∣ ∣. 10. ∣ ∣ ∣∆L c1...cm(i+1) ∣ ∣ ∣ = 2i i + 2 ∣ ∣ ∣∆L c1...cmi1 ∣ ∣ ∣. 11. If a < b and dj(a) = dj(b) for j < m, but dm(a) 6= dm(b), then 1) (a, b] ⊂ ∆L d1(a)...dm−1(a), 2) ∆L d1(a)...dm−1(a)dm(b)(dm+1(b)+1) ⊂ (a, b]. 12. If dm(a) > dm(b), but dj(a) = dj(b) for j < m, then ∆L d1(a)...dm−1(a)dm(b)(dm+1(b)+1) ⊂ (a, b). Definition 3. Let (cn) be a fixed sequence of positive integers and (kn) be a fixed increasing sequence of positive integers. Semicylinder with the base ( k1 k2 . . . kn c1 c2 . . . cn ) is a set ∆k1k2...kn c1c2...cn ≡ {x : x = ∆L d1d2...dk... , dki (x) = ci, i = 1, n}. Jo ur na l A lg eb ra D is cr et e M at h. Yu. Zhykharyeva, M. Pratsiovytyi 149 Lemma 1. Semicylinders have the following properties. 1. ∆12...n c1c2...cn = ∆L c1c2...cn . 2. ∆k1...kn c1...cn = ∆k1 c1 ∩ ∆k2 c2 ∩ . . . ∩ ∆kn cn = ∆k1...km c1...cm ∩ ∆ km+1...kn cm+1...cn . 3. Semicylinder is a union of cylinders of rank kn. 4. Semicylinders ∆k c and ∆m d are metrically independent iff k 6= m. 5. The Lebesgue measure of ∆k1k2...kn c1c2...cn is calculated by formula λ(∆k1k2...kn c1c2...cn ) = n ∏ i=1 1 ci(ci + 1) . Proof. Properties 1–3 follows immediately from the definition of semi- cylinder. It is evident that for k = 1 the set ∆k c is an L-cylinder of 1st rank ∆L c , and according to Property 3 λ(∆1 c) = |∆L c | = 1 c(c + 1) . If k = 2, then by definition of the set ∆k c and properties of cylinders ∆2 c = ⋃ i∈N ∆L ic. So, the Lebesgue measure is equal to λ(∆2 c) = ∞ ∑ i=1 |∆L ic| = 1 c(c + 1) ∞ ∑ i=1 1 i(i + 1) = 1 c(c + 1) . For k = 3, we have ∆3 c = ∞ ⋃ i1=1 ∞ ⋃ i2=1 ∆L i1i2c , λ(∆3 c) = ∞ ∑ i1=1 ∞ ∑ i2=1 |∆L i1i2c | = 1 c(c + 1) ∞ ∑ i1=1 ∞ ∑ i2=1 1 i1(i1 + 1)i2(i2 + 1) = 1 c(c + 1) . For any k the Lebesgue measure of the set ∆k+1 c is defined by equality λ(∆k+1 c ) = ∞ ∑ i1=1 . . . ∞ ∑ ik−1=1 |∆L i1...ikc |. Using Property 7 (basic metric relation) we have λ(∆k+1 c ) = ∞ ∑ i1=1 . . . ∞ ∑ ik=1 |∆L i1...ikc | = 1 c(c + 1) ∞ ∑ i1=1 . . . ∞ ∑ ik=1 |∆L i1...ik | = 1 c(c + 1) . Jo ur na l A lg eb ra D is cr et e M at h. 150 Expansions of numbers in positive Lüroth series The last equality follows from that fact: ∞ ∑ i1=1 . . . ∞ ∑ ik=1 |∆L i1...ik | = 1. For n = 2 ∆k1k2 c1c2 = ∆k1 c1 ⋂ ∆k2 c2 = ⋃ ij∈N j∈{1,2,...,k2−1}\{k1} ∆L i1...ik1−1c1ik1+1...ik2−1c2 , λ(∆k1k2 c1c2 ) = 1 c1(c1 + 1) 1 c2(c2 + 1) ∑ ij∈N j∈{1,2,...,k2−1}\{k1} |∆L i1...ik1−1ik1+1...ik2−1 | = = 1 c1(c1 + 1) 1 c2(c2 + 1) = λ(∆k1 c1 )λ(∆k2 c2 ). The last equality provide metric independence of semicylinders ∆k1 c1 and ∆k2 c2 , i.e., semicylinders ∆k c and ∆m d for k 6= m. If k = m, then ∆k c ∩ ∆m d is an empty set for c 6= d and ∆k c = ∆m d for c = d. Thus λ(∆k c ∩ ∆m d ) 6= λ(∆k c )λ(∆m d ). So, semicylinders are not metrically independent. One can prove Property 5 by induction. Lemma 2. The family of supercylindrical sets (finite or countable unions of cylinders in WL) is an algebra, i.e., closed with respect to finite union and complement class of sets. Proof. It is evident that union of two supercylindrical sets A and A′ is a such set. Let us show that intersection of two supercylindrical sets A and A′ is a supercylindrical set. Let A = ⋃ i Ai, A′ = ⋃ j A′ j , where Ai and A′ j are cylindrical sets. Then A ⋂ A′ = ⋃ i ⋃ j [Ai ⋂ A′ j ]. However, Ai ⋂ A′ j is a cylindrical set. Thus A ⋂ A′ is a supercylindrical set by definition. Now we prove that complement B of supercylindrical set B is a such set. Complement of ∆L c1...cm is a union of sets in the form ∆L s1...sm , where m-tuple (s1 . . . sm) takes all possible combinations of L-symbols except for (c1 . . . cm), i.e., complement of cylinder is a countable union of cylinders of the same rank. It is evident that B1 ∪ B2 = B1 ∩ B2. So, if we take into account that intersection of two supercylindrical sets is a such set, then we have that complement of supercylindrical set is a such set. 3. Set of numbers with given sequence of fixed digits Let (cn) be a fixed sequence of positive integers, (kn) be a fixed increasing sequence of positive integers. We consider the set ∆k1k2...kn... c1c2...cn... ≡ {x : x = ∆L d1d2...dk... , dki (x) = ci, i ∈ N}. Jo ur na l A lg eb ra D is cr et e M at h. Yu. Zhykharyeva, M. Pratsiovytyi 151 Theorem 3. Let gn := kn+1 − kn. 1. If gn = 1 for all n and k1 = 1, then set ∆k1k2...kn... c1c2...cn... consists from one point ∆L c1c2...cn... . If inequality gn > 1 is fulfilled for finitely many n, then this set is countable. If inequality gn > 1 is fulfilled for infinitely many n, then it is a continuum set. 2. Lebesgue measure of the set ∆k1k2...kn... c1c2...cn... is equal to 0. Proof. 1. If gn = 1 starting from some n0, then the set ∆k1k2...kn... c1c2...cn... is countable because only for finite set of the first n0 − 1 positions there exists an alternative for L-symbols from at most countable set. If gn > 1 for infinitely many n, then ∆k1k2...kn... c1c2...cn... is a continuum set, because one can establish one-to-one correspondence f between this set and half-interval (0, 1] by formula f(∆L d1d2...dn... ) = α12−1 + α22−2 + . . . + αn2−n + . . . , where αn = 0 if gn = 1, and αn = 1 if gn > 1. 2. If set ∆k1k2...kn... c1c2...cn... is countable, then its Lebesgue measure is equal to 0 by the properties of the Lebesgue measure. So, it is enough to prove statement 2 if it is a continuum set. Let Fk be a closure of a union of all cylinders of rank k whose interior contains point from the set ∆k1k2...kn... c1c2...cn... . Since Fkn ⊃ Fkn+1 and ∆k1k2...kn... c1c2...cn... = ∞ ⋂ n=1 Fkn , we have λ(∆k1k2...kn... c1c2...cn... ) = lim n→∞ λ(Fkn ) by the continuity from above of the Lebesgue measure. From the basic metric relation it follows that |∆L c1c2...ck+1 | = 1 ck+1(ck+1 + 1) |∆L c1c2...ck |. Thus λ(Fkn+1 ) = ∑ i1∈N,...,ikn+1−1∈N |∆L i1i2...c1...c2...cn...ikn+1−1ckn+1 | = = 1 ckn+1 (ckn+1 + 1) ∑ i1∈N,...,ikn+1−1∈N |∆L i1i2...c1...c2...cn...ikn+1−1 | = = 1 ckn+1 (ckn+1 + 1) λ(Fkn+1−1). From the definition of ∆k1k2...kn... c1c2...cn... it follows Fkn = Fkn+1 = . . . = Fkn+1−1. Thus, λ(Fkn+1 ) = λ(Fkn ) ckn+1 (ckn+1 + 1) = λ(Fk1 ) n+1 ∏ i=2 1 cki (cki + 1) n→∞ −−−→ 0. Jo ur na l A lg eb ra D is cr et e M at h. 152 Expansions of numbers in positive Lüroth series 4. Shift operator for L-representation In the set ZL (0,1] of all L-representations of numbers belonging to (0, 1] we introduce a binary relation of equivalence “to have the same tail” (we denote it by ∼). Definition 4. Two L-representations ∆L α1α2...αn... and ∆L β1β2...βn... have the same tail or they are in relation ∼ if there exist positive integers m and k such that αm+j = βk+j for any j ∈ N . It is evident that ∼ is an equivalence relation (i.e., it is reflexive, symmetric, and transitive) and partitions set where it is defined on the equivalence classes. Any equivalence class is a tail set. Any tail set is determinated uniquely by arbitrary its element (representative). Two numbers x and y have the same tail (or they are in relation ∼), if their L-representations are in relation ∼. We denote it by x ∼ y. Lemma 3. Any tail set is a countable dense in (0, 1] set. Proof. Let H be any equivalence class, and x0 = ∆L c1...ck... be its representa- tive. Then for any positive integer m there exists set Hm of numbers x such that αm+j(x) = αk+j(x0) for any j ∈ N , k = 1, 2, . . .. Set H = ⋃ m∈N Hm is a countable union of countable set. So, it is countable. Since number x belongs to set H independently of any finite number of the first L-symbols, we have that there exits point from H in any cylinder of any rank m. Thus, H is an everywhere dense in (0, 1] set. Corollary. Factor set G ≡ (0, 1]/ ∼ is a continuum set. In the set ZL (0,1] we consider shift operator ϕ for L-symbols defined by equality ϕ(∆L α1α2...αn... ) = ∆L α2α3...αn... . This operator is a function ϕ : (0, 1] → (0, 1]. It is clear that function ϕ has a countable set of invariant points {∆L (c), where c ∈ N}. It is surjective but not injective, because preimages of ∆L c1c2...ck... are points ∆L cc1c2...ck... , where c ∈ N (countable set). Lemma 4. Function ϕ is: 1) decreasing on any cylinder of 1st rank; 2) continuous at any point of cylinder of 1st rank and left-continuous at right endpoint of this interval. Proof. 1. Let us consider two points x1 = ∆L α1α2(x1)...αn(x1)... and x2 = ∆L α1α2(x2)...αn(x2)... belonging to interval ∆L α1 such that x1 < x2. Since Jo ur na l A lg eb ra D is cr et e M at h. Yu. Zhykharyeva, M. Pratsiovytyi 153 αn(ϕ(x)) = αn+1(x) and their L-symbols satisfy conditions (1), we have ϕ(x1) > ϕ(x2), and this proves first statement. 2. Since function ϕ is monotonic and bounded on any cylinder of 1st rank, it has finite right and left limits at any point of this interval. Moreover, it has finite left limit at the right endpoint and finite right limit at the left endpoint. Let x = ∆L α1α2(x)...αn(x)... be any irrational point of int ∆L α1 , and (xk) be any sequence of points xk such that lim k→∞ xk = x. It is easy to prove that lim k→∞ xk = x is equivalent to lim k→∞ mk = ∞, where mk is minimal positive integer such that αmk (xk) 6= αmk (x). In fact, lim k→∞ xk = x is equivalent to the following fact: for any M > 0 there exists mk > M and cylinder ∆L α1α2(x)...αmk (x) of rank mk containing all xk starting from some k0. So, from equalities lim k→∞ xk = x and αn(ϕ(x)) = αn+1(x) it follows that lim k→∞ ϕ(xk) = ϕ(x), and this proves continuity of the function ϕ at the point x. Now let x = ∆L α1α2(x)...αn(x) be any rational point of int ∆L α1 . Let us consider sequence x′ k = ∆L α1α2(x)...αn(x)k converging to x and x′ k < x. It is evident that lim k→∞ ϕ(x′ k) = ϕ(x), i.e., function ϕ is left continuous at point x. Now let us consider sequence x′′ k = ∆L α1α2(x)...(αn(x)−1)1k converging to x and x′′ k > x. It is evident that lim k→∞ ϕ(x′′ k) = ϕ(x), i.e., function ϕ is right continuous at point x. Remark. All points x, ϕn(x), n ∈ N , belong to the same tail set, and x ∼ y iff there exists positive integers k and m such that ϕk(x) = ϕm(y). 5. Sets with restrictions on use of L-symbols Definition 5. A number x is called L-rational if its L-representation has a period (1), i.e., x = ∆L c1c2...cm(1). A number is called L-irrational if it is not L-rational. Any L-rational number is a right endpoint of cylinder, moreover number ∆L c1c2...cm(1) is a right endpoint of ∆L c1c2...cm . Vice versa, right endpoint of any cylinder is L-rational number. It is easy to prove that any L- rational number is rational, but not all rational numbers are L-rational. For example, number ∆(12) is rational, but is not L-rational. Jo ur na l A lg eb ra D is cr et e M at h. 154 Expansions of numbers in positive Lüroth series Theorem 4. The set C ≡ C[L, V ] = {x : x = ∆L d1d2...dn... , dn(x) ∈ V ⊂ N} is 1. a half-interval (0, 1] if V = N ; 2. a nowhere dense non-closed set of zero Lebesgue measure coinciding with its closure with respect to countable set if V 6= N ; 3. self-similar if V is a finite set and N -self-similar if V is an infinite set; moreover, its self-similar (N -self-similar) dimension αs is a solution of equation ∑ v∈V ( 1 v(v + 1) )x = 1 if |V | < ∞; (2) and is a number αs = sup n    x : ∑ v:V ∋v≤n ( 1 v(v + 1) )x = 1    if |V | = ∞. (3) Proof. Statement 1 is evident. 2. Let V 6= N . It is easy to see that C ⊂ ⋃ k∈V ∆L k , C ⊂ ⋃ ki∈V i∈N ∆L k1k2...kn ≡ Fn ⊂ Fn−1, C = ∞ ⋂ k=1 Fk = lim k→∞ Fk. Let (a, b) be any subinterval of (0, 1]. It is evident that cylinder ∆L d1(b)...dm(b)dm+1(b)+1 ⊂ (a, b), where dm(b) 6= dm(a). Let α and β be the endpoints of the cylinder ∆L d1(b)...dm(b)(dm+1(b)+1)v, where v ∈ N\V . Then the interval (α, β) does not contain points of the set C. So, the set C is a nowhere dense set by definition. For Lebesgue measure λ of the set C the following relation holds: λ(C) ≤ ∑ k1∈V . . . ∑ kn∈V |∆L k1...kn | = ∑ k1∈V . . . ∑ kn∈V n ∏ i=1 1 ki(ki + 1) = bn n→∞ −−−→ 0, where 0 < bn = ∑ k∈V 6=N 1 k(k + 1) < 1. So, λ(C) = 0. 3. Since C = ⋃ v∈V [∆L v ∩ C] and 1) C kv∼ ∆L v ∩ C, where k = 1 v(v + 1) , 2) (∆L vi ∩ C) ∩ (∆L vj ∩ C) = ∅, the set C is self-similar if V is finite, and N -self-similar if V is infinite. According to the definition, a self-similar (N -self-similar) dimension is a solution of (2) (or determined by (3) respectively). Jo ur na l A lg eb ra D is cr et e M at h. Yu. Zhykharyeva, M. Pratsiovytyi 155 6. Random variable with independent L-symbols Theorem 5. Random variable ξ = ∆L τ1τ2...τk... with the following distribu- tions of L-symbols τk: P{τk = i} = pik, i ∈ N , has a pure Lebesgue type, moreover, 1. discrete iff M = ∞ ∏ k=1 max i {pik} > 0; 2. absolutely continuous iff S = ∞ ∏ k=1 ( ∞ ∑ i=1 √ pik i(i + 1) ) > 0; (4) 3. singular in other cases, i.e., if M = 0 = S. Proof. Let {(Ωk, Bk, µk)} and {(Ωk, Bk, νk)} be two sequences of prob- ability spaces such that Ωk = N , Bk is a σ-algebra of all subsets of Ωk, µk(i) = pik, νk(i) = 1 i(i + 1) , k ∈ N, where pik is an element of the matrix ‖pik‖ determining the distribution of the random variable ξ. It is evident that measure µk is absolutely continuous with respect to measure νk (µk ≪ νk) for all k ∈ N . Let us consider the infinite products of probability spaces (Ω, B, µ) = ∞ ∏ k=1 (Ωk, Bk, µk), (Ω, B, ν) = ∞ ∏ k=1 (Ωk, Bk, νk). From Kakutani’s theorem [6] it follows that µ ≪ ν iff ∞ ∏ k=1 ∫ Ωk √ dµk dνk dνk > 0, where integral is the Hellinger integral. In this case the last inequality is equivalent to condition (4). Therefore, from the condition (4) it follows that the measure µ is absolutely continu- ous with respect to the measure ν. Let us consider the mapping Ω f −→ [0; 1] defined by equality ∀ω = (ω1, . . . , ωk, . . .) ∈ Ω : f(ω) = ∆L ω1...ωk... . Jo ur na l A lg eb ra D is cr et e M at h. 156 Expansions of numbers in positive Lüroth series For any Borel set E, we define the measures µ∗ and ν∗ as the image measures of µ and ν under mapping f : µ∗(E) = µ(f−1(E)), ν∗(E) = ν(f−1(E)). The measure µ∗ coincides with the probabilistic measure Pξ and the measure ν∗ coincides with the probabilistic measure Pψ, which equivalent to Lebesgue measure λ. From the absolutely continuity of the measure µ with respect to the measure ν it follows that the measure µ∗ is absolutely continuous with respect to the measure ν∗. Since ν∗ ∼ λ, from condition (4) it follows that the random variable ξ is of absolutely continuous distribution. 7. L-representation and fractal analysis of subsets of [0, 1] Definition 6. Hausdorff-Besicovitch dimension of bounded set E ⊂ R1 is a number α0(E) = sup{α : Hα(E) 6= 0} = inf{α : Hα(E) = 0}, where Hα(E) is a α-dimensional Hausdorff measure of E defined by equality Hα(E) = lim ε→0 inf d(Ei)<ε { ∑ i dα(Ei) : E ⊂ ⋃ i Ei } , d(Ei) is a diameter of the set Ei. Let W be a class of sets such that they are unions of L-cylinders of the following form: (1) n ⋃ i=k ∆L c1...cmi , (2) ∞ ⋃ i=k ∆L c1...cmi , where k, n are arbitrary positive integers. It is clear that any cylinder belongs to class W , because for k = 1 set (2) is a cylinder as well as set (1) is a cylinder for k = n. Lemma 5. For any u ≡ (a, b) ⊂ (0, 1] there exists at most 4 sets belonging to class W covering u and having length not exceeding |u|. Proof. The following cases are possible: 1. Numbers a and b belong to different L-cylinders of rank 1; 2. a and b belong to the same L-cylinder of rank 1. Consider every case separately. 1.1. Let a and b belong to neighbouring L-cylinders of 1st rank ∆L d1(b)+1 and ∆L d1(b) respectively, and c = sup ∆L d1(b)+1. Jo ur na l A lg eb ra D is cr et e M at h. Yu. Zhykharyeva, M. Pratsiovytyi 157 a) If a = c (it is equivalent to dj(a) = 1 for j > 1), then for covering u it is enough two sets from W : ∞ ⋃ j=d2(b)+1 ∆L d1(b)j , ∆L d1(b)d2(b), (5) having the length not exceeding b − a (first one belongs to (a, b], second satisfies Property 6 of cylinders). b) If a 6= c, then there exists dk(a) 6= 1. Let us consider the least such k. Then ∆L d1(a)...dk−1(a)1 ⊂ (a, c] and sets dk(a)−1 ⋃ j=1 ∆L d1(a)...dk(a)j and ∆L d1(a)...dk(a) (6) cover (a, c] and have length not exceeding c−a, and therefore, not exceeding b − a. Half-interval (c, b] is covered by two sets (6). So, for covering (a, b] it is enough 4 sets belonging to W . 1.2. If there exists cylinder ∆L m ⊂ (a, b], then (a, b] is covered by the sets ∞ ⋃ j=m ∆L j , ∞ ⋃ j=d2(b)+1 ∆L d1(b)j , ∆L d1(b)d2(b), belonging to W and having length lesser than b − a. 2. Let a and b belong to the same cylinder of 1st rank ∆L d1(b). Then there exists positive integer m such that a and b belong to the same cylinder of rank m, but to different cylinders of rank m + 1: ∆L d1(b)...dm(b)dm+1(a) and ∆L d1(b)...dm(b)dm+1(b). Repeating the same arguments as in the case 1, we obtain the same result: for covering (a, b] it is enough at most four sets belonging to W and having length not exceeding b − a. Theorem 6. For determination of Hausdorff-Besicovitch dimension of any Borel subset of (0, 1] it is enough to use covering by sets belonging to class W . Proof. In fact, if u is an arbitrary half-interval belonging to covering E, then there exists at most 4 sets ω1, ω2, ω3, ω4 belonging to W such that |ωi| α ≤ |uα| for any α > 0. If lαε (E) = inf |vk|≤ε ∑ k |vk| α, Jo ur na l A lg eb ra D is cr et e M at h. 158 Expansions of numbers in positive Lüroth series where E ⊂ ⋃ k vk and vk ∈ W , then mα ε (E) ≤ lαε (E) ≤ 4mα ε (E) for any ε > 0. Therefore Hα(E) ≤ Hα L(E) ≡ lim ε→∞ lαε (E) ≤ 4Hα(E), that is Hα L(E) and Hα(E) simultaneously (with respect to α) take the values 0 and ∞. Consequently, α0(E) = inf{α : Hα L(E)}. Theorem 7. Continuous strictly increasing probability distribution func- tion F of the random variable with independent identically distributed L-symbols preserve the Hausdorff-Besicovitch dimension iff pi = 1 i(i + 1) , ∀ i ∈ N. (7) Proof. If Equality (7) holds, then distribution is uniform on [0, 1], and it is evident that probability distribution function preserve the Hausdorff- Besicovitch dimension. Suppose that there exists pm 6= 1 m(m+1) . Let pm < 1 m(m+1) . Then there exists pc > 1 c(c+1) , i.e., there exist pm and pc such that ( pm − 1 m(m + 1) )( pc − 1 c(c + 1) ) < 0. Then for any a ∈ N , m 6= a 6= c, there exists g ∈ {m, c} such that ( pa − 1 a(a + 1) )( pg − 1 g(g + 1) ) ≥ 0. (8) Let us consider set C ≡ C[L, {a, g}] and its image C ′ = F (C) under transformation F . These sets are self-similar and their self-similar dimen- sions coincides with Hausdorff-Besicovitch dimensions and are solutions of the following equations a−x(a + 1)−x + g−x(g + 1)−x = 1 and pxa + pxg = 1 respectively. However, from (8) and pg 6= g−1(g +1)−1 it follows that their solutions does not coincide. Thus, α0(C) 6= α0(C ′). This contradiction proves the theorem. References [1] S. Albeverio, O. Baranovskyi, M. Pratsiovytyi, and G. Torbin, The Ostrogradsky series and related Cantor-like sets, Acta Arith. 130 (2007), no. 3, 215–230. Jo ur na l A lg eb ra D is cr et e M at h. Yu. Zhykharyeva, M. Pratsiovytyi 159 [2] O. M. Baranovskyi, M. V. Pratsiovytyi, and G. M. 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Ser. 1, Physics & Mathematics (2008), no. 9, 200–211 (in Ukrainian). [18] Yu. I. Zhykharyeva and M. V. Pratsiovytyi, Properties of distribution of the random variable with independent symbols of positive Lüroth series representation, Trans. IAMM NAS Ukraine 23 (2011), 71–83 (in Ukrainian). Jo ur na l A lg eb ra D is cr et e M at h. 160 Expansions of numbers in positive Lüroth series Contact information Yu. Zhykharyeva Physics and Mathematics Institute, Drago- manov National Pedagogical University, Pyro- gova St. 9, 01601 Kyiv, Ukraine E-Mail: july2105@mail.ru M. Pratsiovytyi Physics and Mathematics Institute, Drago- manov National Pedagogical University, Pyro- gova St. 9, 01601 Kyiv, Ukraine E-Mail: prats4@yandex.ru Received by the editors: 02.07.2012 and in final form 02.07.2012. Yu. Zhykharyeva, M. Pratsiovytyi

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