Cantor series expansions and packing dimension faithfulness

TL;DR

This paper develops a general theory of packing measures and dimensions, introducing faithfulness concepts and providing a sharp criterion for the faithfulness of Cantor series expansion-based packing families.

Contribution

It introduces the notion of faithfulness for packing families, establishes packing analogues of Billingsley's theorems, and provides the first sharp condition for faithfulness of Cantor series expansion families.

Findings

Established necessary and sufficient conditions for packing dimension faithfulness.

Developed packing analogues of Billingsley's theorems.

Identified sharp criteria distinguishing faithful and non-faithful packing families.

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.