Randomness and Solovay degrees

Abstract

We consider the behaviour of Schnorr randomness, a randomness notion weaker than Martin-Löf's, for left-r.e. reals under Solovay reducibility. Contrasting with results on Martin-Löf-randomness, we show that Schnorr randomness is not upward closed in the Solovay degrees. Next, some left-r.e. Schnorr random α is the sum of two left-r.e. reals that are far from random. We also show that the left-r.e. reals of effective dimension > r, for some rational r, form a filter in the Solovay degrees.