Intrinsic Diophantine approximation: some new results.

In the 80's, Mahler suggested to study the rational approximation of elements of the middle third Cantor set by rationals lying in the middle third Cantor set. This question raised many interests since and has been widely investigated. In this talk, we will provide extensions of existing results, established in the case of the middle-third Cantor set by Tan, Wang and Wu, to a large class of rational self-similar IFS's. In addition, we will provide new results regarding reduced fractions lying in the middle third Cantor set. These results suggest in particular that the known (lower) bounds for the elements intrinsically psi-approximable of the middle-third Cantor set should be sharp.