In several domains (geophysics, cardiology, finance, and Internet traffic), numerical studies reveal that the fluctuations of characteristic quantities possess scaling invariance properties. These scaling invariance measured on empirical data are reminiscent of properties verified theoretically by mathematical objects, which are called (self-similar) multifractals. A multifractal is a measure, a function or a stochastic process, whose local regularity changes rapidly from one point to another. It is striking how multifractal properties arise in many mathematical fields: dynamical systems, probability, harmonic analysis, and recently Diophantine approximation. The project's main objective is the investigation of the various aspects of the interactions between multifractal theory (more generally, geometric measure theory) and metric number theory. Since the 1990's, such interactions appear to be key issues when performing the regularity analysis of many multifractal models, hence we also plan to develop many mathematical models (functions, measures, stochastic processes) based on the results we will obtain. We will focus on four challenging directions: - Multifractals and Diophantine approximation, Mass transference principles: Recently, interactions between multifractal theory and metric theory of Diophantine approximation have been pointed out, giving new perspectives in both fields. Examples are the heterogeneous Diophantine approximation results, which allow to investigate the approximation of real numbers by rational numbers which are constrained by the fact that their digit frequencies (in any basis) are fixed. We want to push further these interactions, in particular by studying heterogeneous approximation under other natural constraints, and by obtaining various mass transference principles in various contexts. - Large intersection properties: A set E included in R^d belongs to the class of sets with large intersection properties (introduced by K. Falconer) when for any countable family of similarities of R^d, the Hausdorff dimension of the intersection of the sets f_i(E) has same Hausdorff dimension as E. This remarkable property has been proved to hold for many sets arising in metric number theory (in particular, there are many connections with the mass transference principles discussed in the preceding item), and plays an important role when performing the multifractal analysis of many objects. We aim at proving these properties for several sets families relevant in number theory, at introducing a notion of large intersection in spaces other than R^d (for instance inside a Cantor set), and at using our results for the study of the multifractal nature of various objects. - Dynamical Diophantine Approximation: A longstanding issue concerns the equi-distribution properties of the orbit of a point x under the action of a dynamical system (X,T). Recently, there has been some remarkable progress for one of the most natural dynamical system: ([0,1], T) where T is the doubling shift, and the corresponding results on the distribution properties of the orbits rely on multifractal properties. Our goal is to obtain comparable results for more general dynamical systems (like expanding Markov maps, or the Gauss map). This will have consequences on number theory, since the Gauss map is intimately related to the continued fractions. - Development of multifractal models: Since the 1990's, it has been regularly stated that the local regularity properties of many objects (Fourier or wavelet series, Lévy processes, "typical" functions) are related to questions of generalized Diophantine approximation, i.e. the approximation by families of points which are not the rational numbers. We intend to use the results we described above for two purposes: first to study the multifractal nature of classical objects, and then to develop new multifractal models (whose study requires new tools in metric number theory.