Turing degrees of reals of positive effective packing dimension

A relatively longstanding question in algorithmic randomness is jail 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 HaLlSdorff 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. (c) 2008 Elsevier B.V. All rights reserved.