Ubiquity and large intersections properties under digit frequencies constraints

International audienceWe are interested in two properties of real numbers: the first one is the property of being well-approximated by some dense family of real numbers {x(n)}(n >= 1), such as rational numbers and more generally algebraic numbers, and the second one is the property of having given digit frequencies in some b-adic expansion. We combine these two ways of classifying the real numbers, in order to provide a finer classification. We exhibit sets S of points x which are approximated at a given rate by some of the {x(n)}(n) those x(n) being selected according to their digit frequencies. We compute the Hausdorff dimension of any Countable intersection of such sets S, and prove that these sets enjoy the so-called large intersection property