On the $G$-isomorphism of probability and dimensional theories of representations of real numbers and fractal faithfulness of systems of coverings

Garko, Iryna; Nikiforov, Roman; Torbin, Grygoriy

doi:10.1090/tpms/1006

Cited by 3 publications

(3 citation statements)

References 13 publications

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“…Different approaches and special methods for the determination of the Hausdorff dimension are collected in [23,24,31]. A new approach based on the theory of transformations preserving the Hausdorff dimension (DP-transformations) was presented in [7,9] and has been developed in [4,26,27]. In this paper we develop another approach which is deeply connected with the theory of DP-transformations as well as with the following well known approach: to simplify the calculation of the Hausdorff dimension of a given set it is extremely useful to have an appropriate and a relatively narrow family of admissible coverings which lead to the same value of the dimension.…”

Section: Introductionmentioning

confidence: 99%

On the Hausdorff dimension faithfulness and the Cantor series expansion

Albeverio¹,

Ivanenko²,

Lebid³

et al. 2020

Methods Funct. Anal. Topol.

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We study families \Phi of coverings which are faithful for the Hausdorff dimension calculation on a given set E (i. e., special relatively narrow families of coverings leading to the classical Hausdorff dimension of an arbitrary subset of E) and which are natural generalizations of comparable net-coverings. They are shown to be very useful for the determination or estimation of the Hausdorff dimension of sets and probability measures. We give general necessary and sufficient conditions for a covering family to be faithful and new techniques for proving faithfulness/non-faithfulness for the family of cylinders generated by expansions of real numbers. Motivated by applications in the multifractal analysis of infinite Bernoulli convolutions, we study in details the Cantor series expansion and prove necessary and sufficient conditions for the corresponding net-coverings to be faithful. To the best of our knowledge this is the first known sharp condition of the faithfulness for a class of covering families containing both faithful and non-faithful ones.Applying our results, we characterize fine fractal properties of probability measures with independent digits of the Cantor series expansion and show that a class of faithful net-coverings essentially wider that the class of comparable ones. We construct, in particular, rather simple examples of faithful families \scrA of net-coverings which are "extremely non-comparable" to the Hausdorff measure.Ми дослiджуємо сiм'ї \Phi покриттiв, якi є довiрчими для обчислення розмiрностi Хаусдорфа-Безиковича на певнiй множинi E (тобто, спецiальнi вiдносно вузькi сiм'ї покриттiв, яких достатньо для коректного обчислення класичної розмiрностi Хаусдорфа-Безиковича довiльної пiдмножини множини E) i якi є природним узагальненням порiвнянних мережевих покриттiв. В роботi показано, що такi сiм'ї є дуже корисними для обчислення чи оцiнки розмiрностi Хаусдорфа-Безиковича множин та ймовiрнiсних мiр.Нами отримано загальнi необхiднi та достатнi умови довiрчостi для сiмей покриттiв та запропоновано нову технiку доведення довiрчостi/недовiрчостi для сiмей цилiндрiв, породжених рiзними розкладами дiйсних чисел. Маючи додаткову мотивацiю в мультифрактальному аналiзi нескiнченних згорток Бернуллi, ми детально дослiдили розклади Кантора та довели необхiднi та достатнi умови довiрчостi вiдповiдних сiмей покриттiв мережевими цилiндрами. Наскiльки нам вiдомо, цi результати є першими критерiями довiрчостi для класу сiмей покриттiв, що мiстить як довiрчi, так i недовiрчi сiм'ї.Застосовуючи отриманi результати, ми дослiдили тонкi фрактальнi властивостi ймовiрнiсних мiр з незалежними символами розкладiв Кантора i показали, що клас довiрчих мережевих покриттiв суттєво ширше за клас порiвнянних. Ми побудували, зокрема, досить простi приклади довiрчих сiмей \scrA мережевих покриттiв, якi є "екстремально непорiвнянними" вiдносно мiри Хаусдорфа.

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Section: Introductionmentioning

confidence: 99%

On the Hausdorff dimension faithfulness and the Cantor series expansion

Albeverio¹,

Ivanenko²,

Lebid³

et al. 2020

Methods Funct. Anal. Topol.

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“…The fractal and multifractal approaches to the study of such measures are known to be extremely useful (see, e.g., [7,12,39] and references therein). The study of fractal properties of different families of singularly continuous probability measures (see, e.g., [7,16,26,28,27,29,31,38,1,42] and references therein) can be used to solve non-trivial problems in the metric number theory ( [8,9,5,4,10,30,21]), in the theory of dynamical systems and DP-transformations and in fractal analysis ( [6,3,11,19,18,17,22,41,43]).…”

Section: Introductionmentioning

confidence: 99%

On singularity of distribution of random variables with independent symbols of Oppenheim expansions

Sydoruk

Torbin

2017

Modern Stoch. Theory Appl.

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The paper is devoted to the restricted Oppenheim expansion of real numbers (ROE), which includes already known Engel, Sylvester and Lüroth expansions as partial cases. We find conditions under which for almost all (with respect to Lebesgue measure) real numbers from the unit interval their ROE-expansion contain arbitrary digit i only finitely many times. Main results of the paper state the singularity (w.r.t. the Lebesgue measure) of the distribution of a random variable with i.i.d. increments of symbols of the restricted Oppenheim expansion. General non-i.i.d. case is also studied and sufficient conditions for the singularity of the corresponding probability distributions are found.

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“…The fractal and multifractal approaches to the study of such measures are known to be extremely useful (see, e.g., [4,13,34] and references therein). Study of fractal properties of different families of singularly continuous probability measures (see, e.g., [4,18,22,21,23,24,25,35] and references therein) can be used to solve non-trivial problems in metric number theory ( [1,2,7,8,12,26,27]), in the theory of dynamical systems and DP-transformations and in fractal analysis ( [3,9,10,15,16,17,20,36,38]).…”

Section: Introductionmentioning

confidence: 99%

On singularity of distribution of random variables with independent symbols of Oppenheim expansions

Sinelnyk¹,

Torbin²

2017

Preprint

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The paper is devoted to restricted Oppenheim expansion of real numbers (ROE),which includes as partial cases already known Engel, Silvester and Lüroth expansions. We find conditions under which for almost all (with respect to Lebesgue measure) real numbers from the unit interval their ROE-expansion contain arbitrary digit i only finitely many times. Main results of the paper states the singularity (w.r.t. Lebesgue measure) of the distribution of random variable with i.i.d increments of symbols of restricted Oppenheim expansion. General non-i.i.d. case are also studied and sufficient conditions for the singularity of the corresponding probability distributions are found.

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On the $G$-isomorphism of probability and dimensional theories of representations of real numbers and fractal faithfulness of systems of coverings

Cited by 3 publications

References 13 publications

On the Hausdorff dimension faithfulness and the Cantor series expansion

On the Hausdorff dimension faithfulness and the Cantor series expansion

On singularity of distribution of random variables with independent symbols of Oppenheim expansions

On singularity of distribution of random variables with independent symbols of Oppenheim expansions

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