Which DNR can be minimal

TL;DR

This paper improves conditions for the existence of minimal degree diagonally non-recursive functions related to order functions and answers a question about the existence of a specific type of weakly meager covering real with particular computational properties.

Contribution

It refines the conditions under which minimal degree DNR functions exist and constructs a real with specific non-computability and covering properties, answering an open question.

Findings

Reduced the factor in the tree thinning argument from 2^j to j.

Established existence of DNR_h of minimal degree under weaker growth conditions.

Constructed a real that is weakly meager covering, does not compute Schnorr random reals, and does not Schnorr cover REC.

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.