A refinement of the Snu-based multifractal formalism

In this work, we introduce a generalization of the Snu spaces underlying a multifractal formalism for non-concave spectra. We prove that the essential topological properties of the Snu spaces can be transposed in this context; in particular, these new spaces are metric. More importantly, we show that the associated multifractal formalism can detect the logarithmic correction in a Brownian motion resulting from the law of the iterated logarithm. We also build two families of multifractal functions with prescribed pointwise regularity and displaying a logarithmic correction in order to illustrate the usefulness of these generalized spaces.