Cantor series expansions and packing dimension faithfulness

Abstract

The paper is devoted to the development of general theory of packing measures and dimensions via introducing the notion of <<faithfulness of a packing family for $\dim_P$ calculation>> 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.