Partial Randomness and Dimension of Recursively Enumerable Reals

A real α is called recursively enumerable ("r.e." for short) if there exists a computable, increasing sequence of rationals which converges to α. It is known that the randomness of an r.e. real α can be characterized in various ways using each of the notions; program-size complexity, Martin-Lof test, Chaitin Ω number, the domination and Ω-likeness of α, the universality of a computable, increasing sequence of rationals which converges to α, and universal probability. In this paper, we generalize these characterizations of randomness over the notion of partial randomness by parameterizing each of the notions above by a real T C (0,1], where the notion of partial randomness is a stronger representation of the compression rate by means of program-size complexity. As a result, we present ten equivalent characterizations of the partial randomness of an r.e. real. The resultant characterizations of partial randomness are powerful and have many important applications. One of them is to present equivalent characterizations of the dimension of an individual r.e. real. The equivalence between the notion of Hausdorff dimension and compression rate by program-size complexity (or partial randomness) has been established at present by a series of works of many researchers over the last two decades. We present ten equivalent characterizations of the dimension of an individual r.e. real.