Spectal dimension of fractal sets

We consider an optimal partial 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 ϵ and the semi-major axis is δ, we set δ = ϵα, where 0 < α < 1 is an exponent characterizing the anisotropy of the covers. The optimization involves varying the angle of the principal axis to maximize the measure covered by each ellipse. 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 ∼εβ, where β is a generalized dimension. We term β the spectal dimension, because our covering strategy may be used to characterize specular light scattering from fractal sets. We investigate the function β(α) numerically for various sets, showing that it may be different for sets which have the same fractal dimension. (paper)