Representation of families of sets by measures, dimension spectra and Diophantine approximation

A family of sets {F-d}(d) is said to be 'represented by the measure mu' if, for each d, the set F-d comprises those points at which the local dimensio n of mu takes some specific value (depending on d). Finding the Hausdorff d imension of these sets may then be thought of as finding the dimension spec trum, or multifractal spectrum, of mu. This situation pertains surprisingly often, with many familiar families of sets representable by measures which have simple dimension spectra. Examples are given from Diophantine approxi mation, Kleinian groups and hyperbolic dynamical systems.