Two More Characterizations of K-Triviality

We give two new characterizations of K-triviality. We show that if for all Y such that Ω is Y-random, Ω is (Y ⊕ A)-random, then A is K-trivial. The other direction was proved by Stephan and Yu, giving us the first titular characterization of K-triviality and answering a question of Yu. We also prove that if A is K-trivial, then for all Y such that Ω is Y-random, (Y ⊕ A) ≡ LR Y. This answers a question of Merkle. The other direction is immediate, so we have the second characterization of K-triviality. The proof of the first characterization uses a new cupping result. We prove that if A is not LR below B, then for every set X there is a B-random set Y such that X is computable from Y ⊕ A.