We investigate the initial segment complexity of random reals. Let K(σ) denote prefix-free Kolmogorov complexity. A natural measure of the relative randomness of two reals α and β is to compare complexity and . It is well-known that a real α is 1-random iff there is a constant c such that for all n, . We ask the question, what else can be said about the initial segment complexity of random reals. Thus, we study the fine behaviour of for random α. Following work of Downey, Hirschfeldt and LaForte, we say that iff there is a constant such that for all n, . We call the equivalence classes under this measure of relative randomness K-degrees. We give proofs that there is a random real α so that where is Chaitin's random real. One is based upon (unpublished) work of Solovay, and the other exploits a new idea. Further, based on this new idea, we prove there are uncountably many K-degrees of random reals by proving that . As a corollary to the proof we can prove there is no largest K-degree. Finally we prove that if then the initial segment complexities of the natural n- and m-random sets (namely and are different. The techniques introduced in this paper have already found a number of other applications.
Downey is supported by the Marsden Fund for Basic Science in New Zealand. Yu and Ding are supported by NSF of China No.19931020. Additionally, Yu is supported by a postdoctoral fellowship from the New Zealand Institute for Mathematics and its Applications.