members of thin $\Pi_1^0$ classes and generic degrees

TL;DR

This paper explores the relationship between generic degrees and thin $ ext{Pi}_1^0$ classes, showing that 2-generic degrees lack members of such classes, while 1-generic degrees below $ extbf{0}'$ do contain them.

Contribution

It establishes a connection between genericity levels and membership in thin $ ext{Pi}_1^0$ classes, advancing understanding of their Turing degree structure.

Findings

2-generic degrees contain no members of thin $ ext{Pi}_1^0$ classes

All 1-generic degrees below $ extbf{0}'$ contain members of thin $ ext{Pi}_1^0$ classes

Provides new insights into the structure of Turing degrees and thin classes

Abstract

A $\Pi^{0}_{1}$ class $P$ is thin if every $\Pi^{0}_{1}$ subclass $Q$ of $P$ is the intersection of $P$ with some clopen set. In 1993, Cenzer, Downey, Jockusch and Shore initiated the study of Turing degrees of members of thin $\Pi^{0}_{1}$ classes, and proved that degrees containing no members of thin $\Pi^{0}_{1}$ classes can be recursively enumerable, and can be minimal degree below {\bf 0}$'$. In this paper, we work on this topic in terms of genericity, and prove that all 2-generic degrees contain no members of thin $\Pi^{0}_{1}$ classes. In contrast to this, we show that all 1-generic degrees below {\bf 0}$'$ contain members of thin $\Pi^{0}_{1}$ classes.