Randomness and Solovay degrees
Kenshi Miyabe, André Nies, Frank Stephan · Journal of Logic and Analysis · 2018
We consider the behaviour of Schnorr randomness, a randomness notion weaker than Martin-L ö f's, for left-r.e. reals under Solovay reducibility. Contrasting with results on Martin-L ö f-randomenss, 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.