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- W2471089693 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 $1$." @default.
- W2471089693 created "2016-07-22" @default.
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- W2471089693 date "2016-06-17" @default.
- W2471089693 modified "2023-09-27" @default.
- W2471089693 title "THE GAMMA QUESTION FOR MANY-ONE DEGREES" @default.
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