On the Number of Infinite Sequences with Trivial Initial Segment Complexity

The sequences which have trivial prefix-free initial segment complexity are known as K-trivial sets, and form a cumulative hierarchy of length omega. We show that the problem of finding the number of K-trivial sets in the various levels of the hierarchy is Delta(0)(3). This answers a question of Downey/Miller/Yu (see Downey (2010) [7, Section 10.1.4]) which also appears in Nies (2009) [17, Problem 5.2.16]. We also show the same for the hierarchy of the low for K sequences, which are the ones that (when used as oracles) do not give a shorter initial segment complexity compared to the computable oracles. In both cases the classification Delta(0)(3) is sharp