On Work of Barmpalias and Lewis-Pye: A Derivation on the D.C.E. Reals

These are beautiful results that clarify the behavior of random left-c.e. reals. It has long been understood that all random left-c.e. reals are “essentially interchangeable”. One of the key arguments for this heuristic was given by Kučera and Slaman [8], who showed that, up to multiplicative constants, we cannot approximate one random leftc.e. real faster than another (see Lemma 1.1). The convergence of (1) shows more: all approximations to random left-c.e. reals converge in essentially the same way. This not only solidifies our belief that that random left-c.e. reals are interchangeable, but ironically, it gives us a useful way to contrast them. For example, it follows that Bα{Bβ ą 1 if and only if α ́ β is a random left-c.e. real and Bα{Bβ ă 1 if and only if α ́ β is a random right-c.e. real.