$m$-Rigidity and Finite-One Degrees Inside Typical Many-One Degrees

Abstract

In recent work, the notion of $m$-rigidity was introduced as a sufficient condition for the existence of infinite antichains of $1$-degrees inside many-one degrees. Motivated by a recent preprint of Richter, Stephan, and Zhang on finite-one degrees inside many-one degrees, we study the finite-one structure of the many-one degree of an $m$-rigid set. First, combining bi-immunity of $m$-rigid sets with a theorem of Richter, Stephan, and Zhang, we show that for Lebesgue-almost every set $A$, and for a comeager class of sets $A$, the many-one degree $\deg_m(A)$ contains a least finite-one degree. Second, we prove that if $A$ is $m$-rigid, then $\deg_m(A)$ contains infinitely many pairwise incomparable finite-one degrees. More precisely, we construct representatives $B_S \equiv_m A$, indexed by computable sets $S$, such that $T \setminus S$ infinite implies $B_T \not\leq_{fin} B_S$. Third, inside a single finite-one degree we build a strict ascending chain \[ A_{(1)} <_1 A_{(2)} <_1 \cdots \] of $1$-degrees. These results yield almost-sure and comeager partial answers to the first two open problems posed by Richter, Stephan, and Zhang.