On the lower density of Solovay–Zheng–Rettinger degrees

履歴
2025年10月30日：雑誌に投稿，ページ作成

タイトル
On the lower density of Solovay–Zheng–Rettinger degrees

種類
研究論文

出版情報
TBA

要旨
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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