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Non-cupping, measure and computably enumerable splittings



We show that there is a computably enumerable function f (that is, computably approximable from below) that dominates almost all functions, and fW is incomplete for all incomplete computably enumerable sets W. Our main methodology is the LR equivalence relation on reals: ALRB if and only if the notions of A-randomness and B-randomness coincide. We also show that there are c.e. sets that cannot be split into two c.e. sets of the same LR degree. Moreover, a c.e. set is low for random if and only if it computes no c.e. set with this property.



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Non-cupping, measure and computably enumerable splittings



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