Kučera and Gács independently showed that every infinite
sequence is Turing reducible to a Martin-Löf random sequence. We extend
this result to show that every infinite sequence S is
Turing reducible to a Martin-Löf 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. We show that this is the optimal ratio of query bits to
computed bits achievable with Turing reductions. As an application of this
result, we give a new characterization of constructive dimension in terms of
Turing reduction compression ratios.