Given any 1-random set X and any r in , we construct a set of intrinsic density r which is computable from both r and X. For almost all r, this set will be the first known example of an intrinsic density r set which cannot compute any r-Bernoulli random set. To achieve this, we shall formalize the into and within noncomputable coding methods which work well with intrinsic density.
K.Ambos-Spies, Algorithmic randomness revisited, in: Language, Logic and Formalization of Knowledge, Coimbra Lecture and Proceedings of a Symposium Held in Siena, 1997.
C.G.Jockusch, Jr. and P.E.Schupp, Generic computability, Turing degrees, and asymptotic density, Journal of the London Mathematical Society85(2) (2012), 472–490, ISSN 0024-6107. doi:10.1112/jlms/jdr051.
4.
R.G.Downey and D.R.Hirschfeldt, Algorithmic Randomness and Complexity, Theory and Applications of Computability, Springer, 2010.
5.
A.Nies, Computability and Randomness, Oxford Logic Guides, Oxford University Press, 2009.
6.
J.Reimann and T.A.Slaman, Measures and their random reals, ArXiv (2013), arXiv:0802.2705v2.
7.
M.van Lambalgen, The axiomatization of randomness, The Journal of Symbolic Logic55(3) (1990), 1143–1167, ISSN 0022-4812. doi:10.2307/2274480.
8.
Y.Wang, Randomness and complexity, PhD dissertation, Ruprecht-Karls-Universität Heidelberg, 1996, https://webpages.uncc.edu/yonwang/papers/thesis.pdf.
