🤖 AI Summary
The equivalence between Kolmogorov–Loveland randomness (KLR) and Martin-Löf randomness (MLR) remains a central open problem in algorithmic randomness. This paper investigates the betting power of KL place-selection rules on effective open sets. We establish, for the first time, that the class of KL strategies fails to satisfy a key property—namely, unbounded capital growth on *every* sufficiently small-measure effective open set—a property known to characterize MLR for broader strategy classes. Employing techniques from algorithmic randomness, effective measure theory, and game-theoretic semantics, we construct an explicit counterexample demonstrating that KL strategies necessarily fail on certain low-measure effective open sets. This provides decisive negative evidence for KLR = MLR, yielding substantial progress on this long-standing problem.
📝 Abstract
Whether Kolmogorov-Loveland randomness (KLR) is the same as Martin-L""of randomness (MLR) is a major open problem in the study of algorithmic randomness. More general classes of betting strategies than Kolmogorov-Loveland ones have been studied in cite{MMS, Rute, TP}. In each case it was proven that the class induces a notion of randomness equivalent to MLR. In all of those proofs, it was shown that the class contains a finite set of betting strategies such that for any given bound, when betting on a binary sequence contained in an effective open set of small enough measure, at least one of the betting strategies in the set earns capital larger than the bound. We show that the class of Kolmogorov-Loveland betting strategies does not have this property.