Chaitin's Ω as a continuous function - Hölzl, Rupert; Merkle, Wolfgang; Miller, Joseph; Stephan, Frank; Yu, Liang (The journal of symbolic logic, 2019)
We prove that the continuous function - - - that is defined via - - - for all - - - is differentiable exactly at the Martin-Löf random reals with the derivative having value 0; that it is nowhere monotonic; and that - - - is a left-c.e. - - - -complete real having effective Hausdorff dimension - - - .We further investigate the algorithmic properties of - - - . For example, we show that the maximal value of - - - must be random, the minimal value must be Turing complete, and that - - - for every X. We also obtain some machine-dependent results, including that for every - - - , there is a universal machine V such that - - - maps every real X having effective Hausdorff dimension greater than ε to a real of effective Hausdorff dimension 0 with the property that - - - ; and that there is a real X and a universal machine V such that - - - is rational.