Spatial correlations in multifractals

We consider spatial correlations within multifractals or fractal measures μ(x). Correlation functions such as 〈μ(x${)}^m$μ(x+r${)}^n$〉 are argued to scale as (R/a${)}^{y(m}$,n)(r/a${)}^{\mathrm{z}(\mathrm{m}}$,n), where R is the overall radius of the object and a a short cutoff. The exponents y and z are given in terms of the scaling dimensions $D_q$ of the multifractal. We note that the existence of this single scaling form over the full range of r (a≪r≪R) is incompatible with any description of the measure as a superposition of simple fractal sets of localized singularities.