Anisotropic covering of fractal sets

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 εand the semi-major axis is δ, we set δ=ε^α, where 0<α<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 > ~ ε^β, where βis a generalised dimension. We investigate the function β(α) numerically for various sets, showing that it may be different for sets which have the same fractal dimension.

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