Scale-dependent fractal geometry

A generalization of fractal geometry that allows for scale-dependent behavior is described. The resulting framework, termed "scale-dependent fractal" (SDF) geometry, can be used to quantify and model various phenomena whose geometric complexity may depend on scale. Relations are derived between scale distributions and coverage dimensions in the SDF framework, for one- as well as multi-dimensional sets, and applied to level sets arising in turbulence and mixing.