Turing degrees of reals of positive effective packing dimension

Abstract

A relatively longstanding question in algorithmic randomness is Jan Reimann's question whether there is a Turing cone of broken dimension. That is, is there a real A such that { B : B ⩽ T A } contains no 1-random real, yet contains elements of nonzero effective Hausdorff dimension? We show that the answer is affirmative if Hausdorff dimension is replaced by its inner analogue packing dimension. We construct a minimal degree of effective packing dimension 1. This leads us to examine the Turing degrees of reals with positive effective packing dimension. Unlike effective Hausdorff dimension, this is a notion of complexity which is shared by both random and sufficiently generic reals. We provide a characterization of the c.e. array noncomputable degrees in terms of effective packing dimension.