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On Bernoulli Convolutions Generated by Second Ostrogradsky Series and Their Fine Fractal Properties

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On Bernoulli Convolutions Generated by Second Ostrogradsky Series and Their Fine Fractal Properties On Bernoulli Convolutions Generated by Second Ostrogradsky Series and their Fine Fractal Properties Sergio Albeverio, Mykola Pratsiovytyi, Iryna Pratsiovyta, Grygoriy Torbin no. 459 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, Oktober 2009 On Bernoulli convolutions generated by second Ostrogradsky series and their ¯ne fractal properties Sergio Albeverio1;2;3;4 Mykola Pratsiovytyi5;6, Iryna Pratsiovyta7, Grygoriy Torbin8;9 Abstract We study properties of Bernoulli convolutions generated by the second Ostrogradsky series, i.e., probability distributions of random variables X1 (¡1)k+1» » = k ; (1) qk k=1 where qk is a sequence of positive integers with qk+1 ¸ qk(qk + 1), and f»kg are independent random variables taking the values 0 and 1 with probabil- ities p0k and p1k respectively. We prove that » has an anomalously fractal Cantor type singular distribution (dimH (S») = 0) whose Fourier-Stieltjes transform does not tend to zero at in¯nity. We also develop di®erent ap- proaches how to estimate the level of "irregularity" of probability distribu- tions whose spectra are of zero Hausdor® dimension. Using generalizations of the Hausdor® measures and dimensions, ¯ne fractal properties of the probability measure ¹» are studied in details. Conditions for the preserva- tion of the Hausdor®{Billingsley dimension of any subset of the spectrum under its probability distribution function are also obtained. 1 Institut fÄurAngewandte Mathematik, UniversitÄatBonn, Wegelerstr. 6, D- 53115 Bonn (Germany); 2 HCM, SFB-611, IZKS, Bonn; 3BiBoS, Bielefeld{Bonn; 4 CERFIM, Locarno and Acc. Arch., USI (Switzerland); E-mail: albeverio@uni- bonn.de 5 National Pedagogical University, Pyrogova str. 9, 01030 Kyiv (Ukraine) 6 Institute for Mathematics of NASU, Tereshchenkivs'ka str. 3, 01601 Kyiv (Ukraine); E-mail: [email protected] 7 National Pedagogical University, Pyrogova str. 9, 01030 Kyiv (Ukraine); E-mail: [email protected] 8 National Pedagogical University, Pyrogova str. 9, 01030 Kyiv (Ukraine) 9 Institute for Mathematics of NASU, Tereshchenkivs'ka str. 3, 01601 Kyiv (Ukraine); E-mail: [email protected] (corresponding author) 1 Mathematics Subject Classi¯cation (2000): 11K55, 26A30, 28A80, 60E10. Key words: second Ostrogradsky series, Bernoulli convolutions, singularly continuous probability distributions, convolutions of singular measures, Fourier{ Stieltjes transform, Hausdor®{Billingsley dimension, fractals, entropy, transfor- mations preserving fractal dimensions. 1 Introduction About 1861 M. V. Ostrogradsky considered two algorithms for the expansions of positive real numbers in alternating series: X (¡1)k+1 ; where qk 2 N, qk+1 > qk; (2) q1q2 : : : qk k X (¡1)k+1 ; where qk 2 N, qk+1 ¸ qk(qk + 1) (3) qk k (the ¯rst and the second Ostrogradsky series respectively). They were found by E. Ya. Remez among manuscripts and unpublished papers by M. V. Ostrograd- sky [21]. These series give good rational approximations for real numbers. One can prove (see, e.g., [19]) that for any real number x 2 [0; 1] there exists a sequence fqk = qk(x)g such that qk+1 ¸ qk(qk + 1) and 1 1 1 1 X1 (¡1)k+1 x = ¡ + + ::: + + ::: = : (4) q1(x) q2(x) q3(x) qk(x) qk(x) k=1 If x is irrational, then the expansion (8) is unique and it has an in¯nite number of terms. So, (8) establishes one-to-one mapping between the set of all in¯nite increasing sequences of positive integers (q1; q2 : : : ; qk;:::) with qk+1 ¸ qk(qk + 1) and the set of all irrational numbers from the unit interval. In the present paper we study properties of in¯nite Bernoulli convolutions generated by series of the form (3). Let us shortly recall that an in¯nite(general non-symmetric) Bernoulli convolutions with bounded spectra is the distribution P1 P1 of a random series »kbk; where jbkj < 1, and »k are independent ran- k=1 k=1 dom variables taking values 0 and 1 with probabilities p0k and p1k respectively. Measures of this form have been studied since the 1930's from the pure proba- bilistic point of view as well as for their applications in harmonic analysis, in the theory of dynamical systems and in fractal analysis (see, e.g., [18] for details and references). The main purpose of the paper is to study properties of Bernoulli convolutions generated by the second Ostrogradsky series, i.e., the probability distributions of 2 random variables X1 (¡1)k+1» » = k ; (5) qk k=1 where qk is an arbitrary second Ostrogradsky sequence, i.e., a sequence of positive integers with qk+1 ¸ qk(qk + 1), and f»kg is a sequence of independent random variables taking the values 0 and 1 with probabilities p0k and p1k respectively (p0k + p1k = 1). Since there exists a natural one-to-one correspondence between the set of irrational numbers of the unit interval and the set of in¯nite second Ostrogradsky sequences, we have a natural parametrization of symmetric (p0k = 1 2 ) random variables of the form (5) via the set of irrational numbers. We prove that the distribution of » is anomalously fractal, i.e., it is singular w.r.t. Lebesgue measure and its Hausdor® dimension is equal to zero. We also study properties of the Fourier-Stieltjes transform and coe±cients of these mea- sures. In particular, we show that this class of Bernoulli convolutions does not contain any Rajchman measure, i.e., the Fourier-Stieltjes transform of the distri- bution of » does not tend to zero as t tends to in¯nity. Moreover, in most cases the upper limit of the corresponding sequence of the Fourier-Stieltjes coe±cients is equal to 1. Finite as well as in¯nite convolutions of distributions of random variables of the form (5) are also studied in details. In particular, we show that any such ¯nite convolution has an anomalously fractal spectrum. For the limiting case we prove that the in¯nite convolution is of pure type (i.e., it is either purely discretely distributed or absolutely continuously resp. singularly continuously distributed). Moreover, we present necessary and su±cient conditions for the discreteness of this in¯nite convolution measure and show that the Hausdor® dimension of such an in¯nite convolution can vary from 0 to 1. Let ¹ = ¹» be the probability measure corresponding to ». Since the spec- trum S¹ is of zero Hausdor® dimension and, therefore, the Hausdor® dimension of the measure dimH (¹) := inffdimH (E);E 2 B; ¹(E) = 1g is also equal to zero, we conclude that the classical Hausdor® dimension does not reflect the di®erence between the spectrum and other essential supports (see, e.g., [24]) of the singular 0 0 00 00 00 0 measure ¹. Moreover, if p0k = p 2 (0; 1) and p0k = p 2 (0; 1); p 6= p , then the corresponding random variables »0 and »00 are mutually singularly distributed on the common spectrum S¹, and all of them are singularly continuous w.r.t. Lebesgue measure. So, to study ¯ne fractal properties of the distribution of » it is necessary to apply more delicate tools than the ®¡dimensional Hausdor® measure and the corresponding Hausdor® dimension. To this end we consider the h¡Hausdor® measures (see, e.g.,[11]) and the Hausdor®-Billingsley dimen- sions w.r.t. an appropriate probability measure (see, e.g., [7] or Section 6 for details). As an adequate example of such a measure we consider the measure º¤ corresponding to the uniform distribution on the spectrum S¹. We ¯nd necessary and su±cient conditions for ¹ to be absolutely continuous resp. singular w.r.t. 3 º¤, prove a formula for the calculation of the Hausdor®{Billingsley dimension of the spectrum of » w.r.t. the measure º¤, and show that the distribution func- tion of the measure º¤ can be considered as a good choice for the gauge function h(t). Moreover, in the same section we study internally fractal properties of ». In particular, we ¯nd the Hausdor®{Billingsley dimension of the measure ¹ itself. In the last section of the paper we develop a third approach to the study of the level of "irregularity" of probability distributions whose spectra are of zero Hausdor® dimension. We consider the problem of the preservation of the Hausdor®{Billingsley dimension of subsets of the spectrum under the distribution function F». For the case where the elements of the matrix jpikj are bounded away from zero we ¯nd necessary and su±cient conditions for the preservation of the dimension. 2 Expansions of real numbers via the second Ostrogradsky series. De¯nition. Any numerical series of the following form 1 1 1 1 X1 (¡1)k+1 ¡ + + ::: + + ::: = ; (6) q1 q2 q3 qk qk k=1 where fqkg is any sequence of positive integers with qk+1 ¸ qk(qk + 1) (7) is said to be the second Ostrogradsky series. For any irrational number x 2 [0; 1] there exists a unique sequence fqk = qk(x)g such that 1 1 1 1 X1 (¡1)k+1 x = ¡ + + ::: + + ::: = : (8) q1(x) q2(x) q3(x) qk(x) qk(x) k=1 The sequence fqkg can be determined via the following algorithm: 8 > 1 = q x + ¯ (0 · ¯ < x); > 1 1 1 > q = q ¯ + ¯ (0 · ¯ < ¯ ); <> 1 2 1 2 2 1 q q = q ¯ + ¯ (0 · ¯ < ¯ ); 2 1 3 2 3 3 2 (9) > :::::::::::::::::::::::: > > q : : : q q = q ¯ + ¯ (0 · ¯ < ¯ ); :> k 2 1 k+1 k k+1 k+1 k :::::::::::::::::::::::: Let us consider several examples of the second Ostrogradsky series: 4 1 1 1 1 1 1 1) ¡ + ¡ + ¡ + :::: 1 1 ¢ 2 2 ¢ 3 6 ¢ 7 42 ¢ 43 1806 ¢ 1807 Here q1 = 1; and qk+1 = qk(qk + 1); 8k 2 N: 1 1 1 1 1 X1 (¡1)k¡1 2) ¡ + ¡ + ¡ ::: = ; s s3 s7 s15 s31 smk k=1 where 2 · s 2 N; m1 = 1; mk = 2mk¡1 + 1; 1 1 1 1 1 1 3) ¡ + ¡ + ¡ + ::: ; 2 ¢ 3 24 ¢ 3 28 ¢ 38 28 ¢ 332 264 ¢ 364 2256 ¢ 364 1 1 1 1 X1 (¡1)k¡1 4) ¡ + ¡ + ::: = ; 2 7 59 3541 pk k=1 where fpkg is an in¯nite sequence of prime numbers such that p1 = 2; p2 = 7; p3 = 59;:::; pk is the minimal prime number such that pk ¸ pk¡1(pk¡1 + 1): Let us mention some evident properties of denominators of the second Ostro- gradsky series.
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