The arithmetical complexity of dimension and randomness

Abstract

Constructive dimension and constructive strong dimension are effectivizations of the Hausdorff and packing dimensions, respectively. Each infinite binary sequence A is assigned a dimension dim( A ) ∈ [0,1] and a strong dimension Dim( A ) ∈ [0,1]. Let DIM α and DIM α str be the classes of all sequences of dimension α and of strong dimension α, respectively. We show that DIM 0 is properly Π 0 2 , and that for all Δ 0 2 -computable α ∈ (0, 1], DIM α is properly Π 0 3 . To classify the strong dimension classes, we use a more powerful effective Borel hierarchy where a coenumerable predicate is used rather than an enumerable predicate in the definition of the Σ 0 1 level. For all Δ 0 2 -computable α ∈ [0, 1), we show that DIM α str is properly in the Π 0 3 level of this hierarchy. We show that DIM 1 str is properly in the Π 0 2 level of this hierarchy. We also prove that the class of Schnorr random sequences and the class of computably random sequences are properly Π 0 3 .