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Turing degrees of reals of positive effective packing dimension. (English) Zbl 1191.68304
Summary: 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 $$\{B: B \leqslant _T A\}$$ 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.

MSC:
 68Q05 Models of computation (Turing machines, etc.) (MSC2010) 03D28 Other Turing degree structures 03D32 Algorithmic randomness and dimension
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References:
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