arXiv subjects
Publications and source records attributed to Mykola Lebid.
We formulate and prove a shape theorem for a continuous-time continuous-space stochastic growth model under certain general conditions. Similarly to the classical lattice growth models the proof makes use of the subadditive ergodic theorem. A precise expression for the speed of propagation is given in the case of a truncated free branching birth rate.
The paper is devoted to the development of general theory of packing measures and dimensions via introducing the notion of < > and the packing analogues of the Billingsley dimension. To this aim we study equivalent definitions of packing dimension and prove theorems which can be considered as packing analogues of the famous Billingsley's theorems. The main result of the paper gives necessary and sufficient condition for the packing dimension faithfulness of the family of cylinders generated by the Cantor series expansion. To the best of our knowledge this is the first known sharp condition of the packing dimension faithfulness for a class of packing families containing both faithful and non-faithful ones.
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 $\mathcal{A}$ of net-coverings which are "extremely non-comparable" to the Hausdorff measure.