Cupping with random sets

TL;DR

This paper characterizes K-trivial sets in terms of their cuppability properties, establishing precise conditions under which they are not weakly ML-cuppable or ML-cuppable, thereby resolving a question posed by Kučera.

Contribution

It provides a complete characterization of K-trivial sets based on their cuppability, clarifying the relationship between K-triviality and ML-cuppability.

Findings

K-trivial sets are not weakly ML-cuppable

Sets below zero jump are K-trivial iff not ML-cuppable

Answers Kučera's open question on cuppability notions

Abstract

We prove that a set is K-trivial if and only if it is not weakly ML-cuppable. Further, we show that a set below zero jump is K-trivial if and only if it is not ML-cuppable. These results settle a question of Ku\v{c}era, who introduced both cuppability notions.