Computing halting probabilities from other halting probabilities

In this paper we give precise bounds on the redundancy growth rate that is generally required for the computation of an omega number from another omega number. We show that for each ϵ>1, any pair of omega numbers compute each other with redundancy ϵlog⁡n. On the other hand, this is not true for ϵ=1. In fact, we show that for each omega number ΩU there exists another omega number which is not computable from ΩU with redundancy log⁡n. This latter result improves an older result of Frank Stephan.