Intermediate Intrinsic Density and Randomness

Abstract

Given any 1-random set $X$ and any $r\in (0,1)$, we construct a set of intrinsic density $r$ which is computable from $r\oplus X$. 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 {\tt into} and {\tt within} noncomputable coding methods which work well with intrinsic density.

arXiv comment15 pages, Included revisions suggested by Laurent Bienvenu, Denis Hirschfeldt, and an anonymous referee