Pointed computations and Martin-Löf randomnesss

Schnorr showed that a real X is Martin-Löf random if and only if K(X �n) ≥ n − c for some constant c and all n, where K denotes the prefix-free complexity function. Fortnow (unpublished) and Nies, Stephan and Terwijn [NST05] observed that the condition K(X �n) ≥ n−c can be replaced with K(X �rn ) ≥ rn −c, for any fixed increasing computable sequence (rn), in this characterization. The purpose of this note is to establish the following generalisation of this fact. We show that X is Martin-Löf random if and only if ∃c ∀n K(X �rn ) ≥ rn − c, where (rn) is any fixed pointedly X-computable sequence, in the sense that rn is computable from X in a self-delimiting way, so that at most the first rn bits of X are queried in the computation of rn. On the other hand, we also show that there are reals X which are very far from being Martin-Löf random, but for which there exists some X-computable sequence (rn) such that ∀n K(X �rn ) ≥ rn.