Asymptotic density and the coarse computability bound

TL;DR

This paper investigates the concept of coarse computability at various densities, exploring its relationship with Turing reducibility and characterizing the possible densities for c.e. sets, revealing complex interactions between computability and density.

Contribution

It introduces the notion of coarse computability at density, characterizes the densities for c.e. sets, and demonstrates the influence of Turing reducibility and genericity on coarse computability.

Findings

Existence of sets with the same coarse computability density where one is coarsely computable and the other is not.

Characterization of possible densities for c.e. sets as left-$oldsymbol{ m extit{ extbf{ extSigma}}}^0_3$ reals.

If a $oldsymbol{ m extit{ extbf{ extDelta}}}^0_2$ 1-generic set computes a set with density 1, then that set is coarsely computable at density 1.

Abstract

For $r \in [0,1]$ we say that a set $A \subseteq \omega$ is \emph{coarsely computable at density} $r$ if there is a computable set $C$ such that $\{n : C(n) = A(n)\}$ has lower density at least $r$. Let $\gamma(A) = \sup \{r : A \hbox{ is coarsely computable at density } r\}$. We study the interactions of these concepts with Turing reducibility. For example, we show that if $r \in (0,1]$ there are sets $A_0, A_1$ such that $\gamma(A_0) = \gamma(A_1) = r$ where $A_0$ is coarsely computable at density $r$ while $A_1$ is not coarsely computable at density $r$. We show that a real $r \in [0,1]$ is equal to $\gamma(A)$ for some c.e.\ set $A$ if and only if $r$ is left-$\Sigma^0_3$. A surprising result is that if $G$ is a $\Delta^0_2$ $1$-generic set, and $A \leq\sub{T} G$ with $\gamma(A) = 1$, then $A$ is coarsely computable at density $1$.