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.
We study families Φ 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 A of netcoverings which are "extremely non-comparable" to the Hausdorff measure.
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