The Gamma question for many-one degrees

TL;DR

This paper investigates the possible values of the gamma function for many-one degrees, showing it can take any value in [0,1/2] or be 1, contrasting with Turing degrees where only 0, 1/2, and 1 are possible.

Contribution

It establishes the range of the gamma function for many-one degrees as [0,1/2] union {1}, expanding understanding of coarse computability in degree theory.

Findings

Range of Gamma_m is [0,1/2] ∪ {1}

Gamma_T only takes values 0, 1/2, 1

Gamma_m can attain any value in [0,1/2]

Abstract

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 $1$.