## TURING DEGREES OF REALS OF POSITIVE EFFECTIVE PACKING DIMENSION

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### BibTeX

@MISC{Downey_turingdegrees,

author = {Rod Downey and Noam Greenberg},

title = {TURING DEGREES OF REALS OF POSITIVE EFFECTIVE PACKING DIMENSION},

year = {}

}

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### Abstract

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. 1.