Constructive Dimension and Turing Degrees

Abstract

This paper examines the constructive Hausdorff and packing dimensions of Turing degrees. The main result is that every infinite sequence S with constructive Hausdorff dimension dim H(S) and constructive packing dimension dim P(S) is Turing equivalent to a sequence R with dim H(R)≥(dim H(S)/dim P(S))−ε, for arbitrary ε>0. Furthermore, if dim P(S)>0, then dim P(R)≥1−ε. The reduction thus serves as a randomness extractor that increases the algorithmic randomness of S, as measured by constructive dimension.A number of applications of this result shed new light on the constructive dimensions of Turing degrees. A lower bound of dim H(S)/dim P(S) is shown to hold for the Turing degree of any sequence S. A new proof is given of a previously-known zero-one law for the constructive packing dimension of Turing degrees. It is also shown that, for any regular sequence S (that is, dim H(S)=dim P(S)) such that dim H(S)>0, the Turing degree of S has constructive Hausdorff and packing dimension equal to 1.Finally, it is shown that no single Turing reduction can be a universal constructive Hausdorff dimension extractor, and that bounded Turing reductions cannot extract constructive Hausdorff dimension. We also exhibit sequences on which weak truth-table and bounded Turing reductions differ in their ability to extract dimension.