Two More Characterizations of K-Triviality

Abstract

We give two new characterizations of $K$-triviality. We show that if for all $Y$ such that $\Omega$ is $Y$-random, $\Omega$ is $\left(Y\oplus A\right)$-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 $\Omega$ is $Y$-random, $\left(Y\oplus A\right){\equiv }_{LR}Y$. This answers a question of Merkle and Yu. 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 $A{\nleqq }_{LR}B$, then for every set $X$ there is a $B$-random set $Y$ such that $X$ is computable from $Y\oplus A$.