The Second Generalization of the Hausdorff Dimension Theorem for Random Fractals

TL;DR

This paper extends the understanding of the cardinality of random fractals, showing that for given Hausdorff dimension and Lebesgue measure, there are aleph-two such fractals with specific dimensional properties, advancing the mathematical theory of fractals.

Contribution

It provides a second partial solution to the problem of counting random virtual fractals with fixed Hausdorff dimension and measure, generalizing previous deterministic results.

Findings

Existence of aleph-two random fractals with specified properties

Similar results for other fractal dimensions

Progress on the cardinality problem for random fractals

Abstract

In this paper, we present a second partial solution for the problem of cardinality calculation of the set of fractals for its subcategory of the random virtual ones. Consistent with the deterministic case, we show that for the given quantities of Hausdorff dimension and Lebesgue measure, there are aleph-two virtual random fractals with almost surely Hausdorff dimension of a bivariate function of them and the expected Lebesgue measure equal to the later one. The associated results for three other fractal dimensions are similar to the case given for the Hausdorff dimension. The problem remains unsolved for the case of non-Euclidean abstract fractal spaces.