Computational Randomness and Lowness

We prove that there are uncountably many sets that are low for the class of Schnorr random reals. We give a purely recursion theoretic characterization of these sets and show that they all have Turing degree incomparable to 0 0. This contrasts with a result of Ku cera and Terwijn 5] on sets that are low for the class of Martin-LL of random reals. The Cantor space 2 ! is the set of innnite binary sequences; these are called reals and are identiied with subsets of !. If 2 2 <! , that is, is a nite binary sequence, we denote by ] the set of reals that extend. These form a basis of clopen sets for the usual discrete topology on 2 !. Write jj for the length of 2 2 <!. The Lebesgue measure on 2 ! is deened by stipulating that ] = 2 ?jj. With every set U 2 <! we associate the open set S 2U ]. When it is convenient, we confuse U with the open set associated to it, in particular we write U for the measure of the open set corresponding to U. We use the following abbreviation for the measure conditioned to :