  Kucera and Gacs independently showed that every infinite sequence is Turing
reducible to a Martin-Lof random sequence. This result is extended by showing
that every infinite sequence S is Turing reducible to a Martin-Lof random
sequence R such that the asymptotic number of bits of R needed to compute n
bits of S, divided by n, is precisely the constructive dimension of S. It is
shown that this is the optimal ratio of query bits to computed bits achievable
with Turing reductions. As an application of this result, a new
characterization of constructive dimension is given in terms of Turing
reduction compression ratios.
