A class of random Cantor sets

TL;DR

This paper investigates a class of random Cantor sets, analyzing their fractal dimensions, topological properties, measure-theoretic dimensions, and hitting probabilities, providing comprehensive insights into their geometric and probabilistic structure.

Contribution

It introduces a new class of random Cantor sets and computes their various fractal, topological, and measure-theoretic dimensions, along with hitting probability analysis.

Findings

Determined almost sure Hausdorff, packing, box, and Assouad dimensions.

Computed typical dimensions in the Baire category sense.

Analyzed hitting probabilities for a subclass of these sets.

Abstract

In this paper we study a class of random Cantor sets. We determine their almost sure Hausdorff, packing, box, and Assouad dimensions. From a topological point of view, we also compute their typical dimensions in the sense of Baire category. For the natural random measures on these random Cantor sets, we consider their almost sure lower and upper local dimensions. In the end we study the hitting probabilities of a special subclass of these random Cantor sets.