Left computably enumerable reals and initial segment complexity

We are interested in the computability between left c.e. reals $\alpha$ and their initial segments. We show that the quantity $C(C({\alpha }_n)\mathrm{\mid }{\alpha }_n)$ plays a crucial role in this and in their completeness. We look in particular at Chaitin’s theorem and its relativisation due to Frank Stephan.