Anisotropic covering of fractal sets

TL;DR

This paper explores optimal anisotropic ellipsoidal coverings of fractal sets in 2D, revealing how the covering properties depend on anisotropy and can differ among sets with the same fractal dimension.

Contribution

It introduces a novel method for covering fractals with anisotropic ellipses and analyzes the relationship between anisotropy and covering efficiency.

Findings

The number of points covered scales with epsilon as <N> ~ epsilon^beta.

The function beta(alpha) varies for different fractal sets.

Anisotropic covers reveal differences not captured by fractal dimension alone.

Abstract

We consider the optimal covering of fractal sets in a two-dimensional space using ellipses which become increasingly anisotropic as their size is reduced. If the semi-minor axis is \epsilon and the semi-major axis is \delta, we set \delta=\epsilon^\alpha, where 0<\alpha<1 is an exponent characterising the anisotropy of the covers. For point set fractals, in most cases we find that the number of points N which can be covered by an ellipse centred on any given point has expectation value < N > ~ \epsilon^\beta, where \beta is a generalised dimension. We investigate the function \beta(\alpha) numerically for various sets, showing that it may be different for sets which have the same fractal dimension.