Computing from Projections of Random Points: a Dense Hierarchy of Subideals of the K-trivial Degrees

We study the sets that are computable from both halves of some (Martin-Löf) random sequence, which we call 1{2-bases. We show that the collection of such sets forms an ideal in the Turing degrees that is generated by its c.e. elements. It is a proper subideal of the K-trivial sets. We characterise 1{2-bases as the sets computable from both halves of Chaitin’s Ω, and as the sets that obey the cost function cpx, sq “ ? Ωs ́ Ωx. Generalising these results yields a dense hierarchy of subideals in the Ktrivial degrees: For k ă n, let Bk{n be the collection of sets that are below any k out of n columns of some random sequence. As before, this is an ideal generated by its c.e. elements and the random sequence in the definition can always be taken to be Ω. Furthermore, the corresponding cost function characterisation reveals that Bk{n is independent of the particular representation of the rational k{n, and that Bp is properly contained in Bq for rational numbers p ă q. These results are proved using a generalisation of the Loomis–Whitney inequality, which bounds the measure of an open set in terms of the measures of its projections. The generality allows us to analyse arbitrary families of orthogonal projections. As it turns out, these do not give us new subideals of the K-trivial sets; we can calculate from the family which Bp it characterises. We finish by studying the union of Bp for p ă 1; we prove that this ideal consists of the sets that are robustly computable from some random sequence. This class was previously studied by Hirschfeldt, Jockusch, Kuyper, and Schupp [25], who showed that it is a proper subclass of the K-trivial sets. We prove that all such sets are robustly computable from Ω, and that they form a proper subideal of the sets computable from every (weakly) LR-hard random sequence. We also show that the ideal cannot be characterised by a cost function, giving the first such example of a Σ3 subideal of the K-trivial sets.