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30 Oct 2025: Submitted to a journal, page created
Title
On the lower density of Solovay–Zheng–Rettinger degrees
Type
Research article
Publication
TBA
Abstract
In the theory of algorithmic randomness, it is well known that the Solovay degrees of left-c.e.\ reals are dense.
In this paper, we establish a corresponding lower density result for degrees of the modified version of Solovay reducibility introduced by Zheng and Rettinger.
It is known that the modified reducibility behaves better for computably approximable reals than the orignal reducibility. We call the modified one Solovay–Zheng–Rettinger reducibility (abbreviated as Solovay–ZR reducibility).
Our proof employs a completely different strategy from the known argument.
Furthermore, we demonstrate the existence of a quasi-minimal Solovay–ZR degree: a weakly computable real such that every left-c.e.\ real Solovay–ZR-reducible to it is necessarily computable.
Finally, we point out that this notion can be regarded as the dual counterpart to variation randomness.
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TBA