Box counting dimensions of generalised fractal nests

TL;DR

This paper generalizes the concept of fractal nests, computes their box counting dimensions, introduces new classes of such nests, and explores their numerical and theoretical properties, highlighting differences from Hausdorff dimensions.

Contribution

It provides formulas for box counting dimensions of generalized fractal nests and introduces novel parameterized classes, advancing understanding of their dimensional properties.

Findings

Formulas for box counting dimensions of generalized nests

Introduction of new parameterized fractal nest classes

Numerical methods to estimate these dimensions

Abstract

Fractal nests are sets defined as unions of unit $n$-spheres scaled by a sequence of $k^{-\alpha}$ for some $\alpha>0$. In this article we generalise the concept to subsets of such spheres and find the formulas for their box counting dimensions. We introduce some novel classes of parameterised fractal nests and apply these results to compute the dimensions with respect to these parameters. We also show that these dimensions can be seen numerically. These results motivate further research that may explain the unintuitive behaviour of box counting dimensions for nest-type fractals, and in general the class of sets where the box-counting dimension differs from the Hausdorff dimension.