Randomness versus superspeedability

A real number is left-computable if it can be effectively approximated from below. A real number is speedable if it has such approximations that can be accelerated infinitely often by a constant factor, and it is superspeedable if it is uniformly speedable by arbitrary factors. We answer a question of Barmpalias by separating the speedable and the superspeedable real numbers. The two latter types of real numbers integrate themselves into a hierarchy of subclasses of the left-computable real numbers that has been studied in a growing body of recent work. We add a new perspective to these studies by juxtaposing this hierarchy with the well-studied hierarchy of algorithmic randomness notions and show that there is a superspeedable real number that is Schnorr random.