Box-counting dimensions of generalised fractal nests

摘要

Fractal nests are sets defined as unions of unitn-spheres scaled by a sequence ofk?αfor someα?>?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.