The arithmetical complexity of dimension and randomness

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 DIMα and DIMαstr be the classes of all sequences of dimension α and of strong dimension α, respectively. We show that DIM0 is properly Π02, and that for all Δ02-computable α ∈ (0, 1], DIMα is properly Π03. 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 Σ01 level. For all Δ02-computable α ∈ [0, 1), we show that DIMαstr is properly in the Π03 level of this hierarchy. We show that DIM1str is properly in the Π02 level of this hierarchy. We also prove that the class of Schnorr random sequences and the class of computably random sequences are properly Π03.