Relativizations of randomness and genericity notions

A set A is a base for Schnorr randomness if it is Turing reducible to a set R that is Schnorr random relative to A , and the notion of a base for weak 1‐genericity can be defined similarly. We show that A is a base for Schnorr randomness if and only if A is a base for weak 1‐genericity if and only if the halting set K is not Turing reducible to A . Furthermore, we define a set A to be high for Schnorr randomness versus Martin‐Löf randomness if and only if every set that is Schnorr random relative to A is also Martin‐Löf random unrelativized, and we show that A is high for Schnorr randomness versus Martin‐Löf randomness if and only if K is Turing reducible to A . Results concerning highness for other pairs of randomness notions are also presented.