CHAITIN’S Ω AS A CONTINUOUS FUNCTION

We prove that the continuous function Ω̂ : 2 → R that is defined via X 7→ ∑ n 2 −K(X n) for all X ∈ 2 is differentiable exactly at the Martin-Löf random reals with the derivative having value 0; that it is nowhere monotonic; and that ∫ 1 0 Ω̂(X) dX is a left-c.e. wtt-complete real having effective Hausdorff dimension 1/2. We further investigate the algorithmic properties of Ω̂. For example, we show that the maximal value of Ω̂ must be random, the minimal value must be Turing complete, and that Ω̂(X)⊕X ≥T ∅′ for every X. We also obtain some machine-dependent results, including that for every ε > 0, there is a universal machine V such that Ω̂V maps every real X having effective Hausdorff dimension greater than ε to a real of effective Hausdorff dimension 0 with the property that X ≤tt Ω̂V (X); and that there is a real X and a universal machine V such that ΩV (X) is rational.