Which DNR can be minimal

Abstract

Khan and Miller proved that for every computable non decreasing unbounded function $h\in \omega^\omega$ (henceforth order function), if $h$ is sufficiently large, then there exists a $DNR_h$ that is of minimal degree. Where $h$ has to satisfy $\lim_{n\rightarrow\infty} h(n)/(2^{k\cdot \prod_{m<n}h(m)})=\infty$ for all $k>0$. Their core argument is that we can thin the tree by a factor of $2^j$ to make $j$ Turing functional split. We improve their result by reducing this factor to $j$. Thus we show that for every order function $h$ with $\lim_{n\rightarrow\infty} h(n)/( \prod_{m<n}h(m))^k=\infty$ for all $k>0$, there exists a $DNR_h$ of minimal degree. We answer a question of Brendle, Brooke-Taylor, Ng and Nies by showing that there exists a $G\in \omega^\omega$ such that $G$ is weakly meager covering, $G$ does not compute any Schnorr random real and $G$ does not Schnorr cover REC.