Intermediate Intrinsic Density and Randomness

TL;DR

This paper constructs sets with any given intrinsic density computable from a random set and a real number, revealing new limitations on their computational power relative to Bernoulli randomness.

Contribution

It introduces novel coding methods to construct sets with prescribed intrinsic density that cannot compute certain Bernoulli random sets.

Findings

Constructed sets with any intrinsic density from random sets and real numbers.

Most such sets cannot compute any r-Bernoulli random set.

Developed formal noncomputable coding techniques for intrinsic density.

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.