Classifying Cantor Sets by their Fractal Dimensions

TL;DR

This paper classifies Cantor sets generated by monotone sequences based on their fractal dimensions, specifically h-Hausdorff and h-Packing measures, linking these classifications to the sequences defining the sets.

Contribution

It provides a new classification framework for Cantor sets using generalized fractal measures and characterizes this classification through the sequences defining the sets.

Findings

Classification of Cantor sets via h-Hausdorff and h-Packing measures

Characterization of classifications in terms of underlying sequences

Extension of Besicovitch and Taylor's work on Cantor sets

Abstract

In this article we study Cantor sets defined by monotone sequences, in the sense of Besicovitch and Taylor. We classify these Cantor sets in terms of their h-Hausdorff and h-Packing measures, for the family of dimension functions h, and characterize this classification in terms of the underlying sequences.