Asymptotic density and the coarse computability bound

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 ${n : C(n) = A(n)}$haslowerdensityatleast$r$\mathrm{.}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 $r \in (0,1]$therearesets$A_0, A_1$suchthat$\gamma(A_0) = \gamma(A_1) = r$where$A_0$iscoarselycomputableatdensity$r$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-${\mathrm{\Sigma }}_3^0$. A surprising result is that if $G$ is a ${\mathrm{\Delta }}_2^0$ $1$-generic set, and $A \leq\sub{T} G$with$\gamma(A) = 1$,then$A$iscoarselycomputableatdensity$1$.