Effective Packing Dimension and Traceability Rod Downey and Keng

The concern of this paper is with effective packing dimension. This concept can be traced back to the work of Borel and Lebesgue who studied measure as a way of specifying the size of sets. Carathéodory later generalized Lebesgue measure to the n-dimensional Euclidean space, and this was taken further by Hausdorff [Hau19] who generalized the notion of s-dimensional measure to include non-integer values for s in any metric space. In the Cantor space with the clopen topology, this can be viewed as a scaling of the usual Lebesgue measure by a factor of s, in the sense of μs([σ]) = 2−s|σ|, where [σ] is the clopen set generated by σ, and 0 ≤ s ≤ 1. This gave rise to the concept of classical Hausdorff dimension, which provided a way of classifying different sets of measure zero, based on the intuition that not all null sets are created equal. There appeared many other related classical notions of fractional dimensions, such as box-counting dimension and packing dimension. The study of effective notions of randomness and their relationship with the Turing degrees was initiated by the early work of de Leeuw, Moore, Shannon and Shapiro [dLMSS56]. The effective versions of these various notions of fractional dimensions have been studied in connection with randomness. The best known examples of such effective notions are the effective Hausdorff, and effective packing dimensions. Hausdorff measure talks about covering the set by open balls from the exterior, while packing measure considers filling up a set from the interior. One can effectivize these two notions by looking at covering with Σ1 open sets in the Cantor space with s-measure. This work took a new direction when various authors Lutz [Lut90, Lut03], Staiger [Sta93], Mayordomo [May02], Artheya et al [AHLM04], and Reimann [Rei04] showed that there were simple characterizations of effective Hausdorff and packing dimensions using Kolmogorov complexity. Indeed, the effective Hausdorff dimension of a real A can be written as