Surjective Functions on Computably Growing Cantor Sets

Every in nite binary sequence is Turing reducible to a random one. This is a corollary of a result of P eter G acs stating that for every co-r.e. closed set with positive measure of in nite sequences there exists a computable mapping which maps a subset of the set onto the whole space of in nite sequences. Cristian Calude asked whether in this result one can replace the positive measure condition by a weaker condition not involving the measure. We show that this is indeed possible: it is su cient to demand that the co-r.e. closed set contains a computably growing Cantor set. Furthermore, in the case of a set with positive measure we construct a surjective computable map which is more e ective than the map constructed by G acs.