I. Stern
SAO
Fractals and multifractals are used to model hierarchical, inhomogenous structures in several areas of astrophysics, notably the distribution of matter at various scales in the universe. It is known however, that current analysis techniques used to assert fractality or multifractality and extract scaling exponents from astrophysical data, have significant limitations and caveats. It is pointed out in this work that some of the difficulties regarding the convergence of algorithms commonly used to determine fractal dimensions or the multifractal scaling spectum, are intrinsically related to 'higher order' structural properties of a fractal, like its 'texture' or 'lacunarity'. The importance of lacunarity for fractal modeling in astrophysics was recently stressed in the literature. In the present work a novel approach to characterize lacunarity based on the formalism of regular variation (in the sense of Karamata) is proposed. It is shown that this approach allows to derive more precise bounds on convergence rates for algorithms which use cover functions to extract scaling exponents ('dimensions'). It also provides a framework to more precisely characterize fractal-like objects with scale dependent properties which may be important for astrophysical applications. An application of the regular variation based fractal modeling to apollonian tiling is also presented, which can be useful to model the morphology of cosmic voids.