A real of strictly positive effective packing dimension that does not compute a real of effective packing dimension one

Abstract

Recently, the Dimension Problem for effective Hausdorff dimension was solved by J. Miller in [14], where the author constructs a Turing degree of non-integral Hausdorff dimension. In this article we settle the Dimension Problem for effective packing dimension by constructing a real of strictly positive effective packing dimension that does not compute a real of effective packing dimension one (on the other hand, it is known via [10. 3. 7] that every real of strictly positive effective Hausdorff dimension computes reals whose effective packing dimensions are arbitrarily close to, but not necessarily equal to, one).

Computability and Randomness

“…There are universal prefix-free machines, and we can take such a machine and define the prefix-free Kolmogorov complex-ity of as ( ) = ( ). The roots of this notion can be found in the work of Levin, Chaitin, and Schnorr, and in a certain sense-like the notion of Kolmogorov complexity more generally-even earlier in that of Solomonoff (see [9,18]). As shown by Schnorr (see Chaitin (1975)), it is indeed the case that is Martin-Löf random if and only if ( ↾ ) ≥ − (1).…”

“…Can a source of partial (algorithmic) randomness be amplified into a source that is fully random, or at least more random? The books Downey and Hirschfeldt [9] and Nies [24] cover material along these lines up to about 2010.…”

“…Hence, instead of having to analyze the relevant distribution and its statistics, we can simply argue about the behavior of the process on a single input. For instance, the 1 Due to space limitations, we omit historical citations and those found in the books Downey and Hirschfeldt [9], Li and Vitányi [18], or Nies [24] from the list of references. In some sections below, we also cite secondary sources where additional references can be found.…”

“…There are now more than a dozen other characterizations of -triviality. Some appear in [9,24], and several others have emerged more recently. These have been used to solve several problems in algorithmic randomness and related areas.…”

Optimal bounds for single-source Kolmogorov extractors

“…(Bienvenu et al [BDS09] independently obtained the first part of the theorem with a more direct proof, but with a reduction from X to Y that is not even guaranteed to be wtt). Conidis [Con12] showed that Fortnow et al's theorem cannot be strengthened to Dim(Y ) = 1, even for Turing reductions.…”