How far is Kolmogorov-Loveland randomness from Martin-Löf randomness?

Abstract

We'll call a map from basic to clopen sets of binary sequences an \textit{encoding} if it preserves uniform Lebesgue measure and the disjointedness relation. For some notion of randomness $R$ we can ask if there is a computable encoding $E$ such that for every non-Martin-Löf-random (non-MLR) sequence $α$, the sequences in $E(α)$ are non-$R$. Thus we can use $E$ and randomness test for $R$ to construct a randomness test for MLR. We show that there is no such $E$ for partial computable randomness (PCR), but there is one for Kolmogorov-Loveland randomness. Indeed, there is a total computable $E$ and a pair of total computable non-monotonic betting strategies such that on every sequence in $E(α)$ at least one of the strategies wins whenever $α$ is non-MLR.