Density-1-bounding and quasiminimality in the generic degrees

TL;DR

This paper investigates whether all nonzero generic degrees are density-1-bounding, exploring techniques to prove this and examining the properties of quasiminimal degrees, with implications for the structure of generic degrees.

Contribution

It introduces new examples of quasiminimal sets that are not density-1 and extends results to uniform and nonuniform degrees, advancing understanding of generic degree structure.

Findings

Many assumptions suffice to prove density-1-bounding.

Constructed examples of non-density-1 quasiminimal sets.

Characterized the randomness level needed for quasiminimality.

Abstract

We consider the question "Is every nonzero generic degree a density-1-bounding generic degree?" By previous results \cite{I2} either resolution of this question would answer an open question concerning the structure of the generic degrees: A positive result would prove that there are no minimal generic degrees, and a negative result would prove that there exist minimal pairs in the generic degrees. We consider several techniques for showing that the answer might be positive, and use those techniques to prove that a wide class of assumptions is sufficient to prove density-1-bounding. We also consider a historic difficulty in constructing a potential counterexample: By previous results \cite{I1} any generic degree that is not density-1-bounding must be quasiminimal, so in particular, any construction of a non-density-1-bounding generic degree must use a method that is able to construct a quasiminimal generic degree. However, all previously known examples of quasiminimal sets are also density-1, and so trivially density-1-bounding. We provide several examples of non-density-1 sets that are quasiminimal. Using cofinite and mod-finite reducibility, we extend our results to the uniform coarse degrees, and to the nonuniform generic degrees. We define all of the above terms, and we provide independent motivation for the study of each of them. Combined with a concurrently written paper of Hirschfeldt, Jockusch, Kuyper, and Schupp \cite{HJKS}, this paper provides a characterization of the level of randomness required to ensure quasiminimality in the uniform and nonuniform coarse and generic degrees.