  A set $A$ is coarsely computable with density $r \in [0,1]$ if there is an
algorithm for deciding membership in $A$ which always gives a (possibly
incorrect) answer, and which gives a correct answer with density at least $r$.
To any Turing degree $\mathbf{a}$ we can assign a value $\Gamma_T(\mathbf{a})$:
the minimum, over all sets $A$ in $\mathbf{a}$, of the highest density at which
$A$ is coarsely computable. The closer $\Gamma_T(\mathbf{a})$ is to $1$, the
closer $\mathbf{a}$ is to being computable. Andrews, Cai, Diamondstone,
Jockush, and Lempp noted that $\Gamma_T$ can take on the values $0$, $1/2$, and
$1$, but not any values in strictly between $1/2$ and $1$. They asked whether
the value of $\Gamma_T$ can be strictly between $0$ and $1/2$. This is the
Gamma question.
  Replacing Turing degrees by many-one degrees, we get an analogous question,
and the same arguments show that $\Gamma_m$ can take on the values $0$, $1/2$,
and $1$, but not any values strictly between $1/2$ and $1$. We will show that
for any $r \in [0,1/2]$, there is an $m$-degree $\mathbf{a}$ with
$\Gamma_m(\mathbf{a}) = r$. Thus the range of $\Gamma_m$ is $[0,1/2] \cup
\{1\}$.
  Benoit Monin has recently announced a solution to the Gamma question for
Turing degrees. Interestingly, his solution gives the opposite answer: the only
possible values of $\Gamma_T$ are $0$, $1/2$, and $1$.
