Distortion and distribution of sets under inner functions. It is a classical result that Lebesgue measure on the unit circle is invariant under inner functions xing the origin and in this setting, the distortion of Hausdor contents has also been studied by Fernández and Pestana. similar results focusing on inner functions with xed points on the unit circle. We will present In particular, our results yield information not only on the size of preimages of sets under inner functions, but also on their distribution with respect to a given boundary point. Join work with Matteo Levi and Oí Soler iGibert.