S2a-reducibility and differentiation in Martin-L\"of random reals

Abstract

Solovay reducibility is studied intensively as a tool to compare the approximability and the degree of randomness of left-c.e. reals. By definition, a real is left-c.e. if it has a left-c.e. approximation, that is, it is the limit of an effective nondecreasing sequence of rationals. If reals $\alpha$ and $\beta$ have left-c.e. approximations $a_0, a_1, \ldots$ and $b_0, b_1, \ldots$, respectively, such that the approximation ratios \[ \frac{\alpha-a_n}{\beta-b_n} \] are bounded from above by a constant, the real $\alpha$ is Solovay reducible to $\beta$. The latter is the case for any such $\alpha$ and $\beta$ and their left-c.e. approximations whenever $\beta$ is Martin-L\"of random by the Ku\v{c}era-Slaman Theorem [DOI:10.1137/S0097539799357441]. This result was substantially strengthened by Barmpalias and Lewis-Pye [DOI:10.1016/j.jcss.2017.06.002], who demonstrated that, under the given assumptions, the approximation ratios are not only bounded but actually converge to a limit, which does not depend on the considered left-c.e. approximations. There is a quest for a suitable extension of Solovay reducibility to the class of all reals. Promising candidates include S2a-reducibility on the set of computably approximable reals by Zheng and Rettinger [DOI:10.1007/978-3-540-27798-9_39] and monotone Solovay reducibility by Titov [DOI:10.1007/978-3-031-95908-0_33]. For the latter, Titov [DOI:10.1017/jsl.2025.10157] demonstrated that the theorems of Ku\v{c}era and Slaman and of Barmpalias and Lewis-Pye extend to all reals. He conjectured further [DOI:10.1017/jsl.2025.10157, Conjecture 3.2] that similar extensions hold for S2a-reducibility in terms of its functional characterization by Kumabe, Miyabe, and Suzuki [DOI:10.3233/COM-230486]. In this work, we refute this conjecture by proving that the analogue of the Barmpalias-Lewis-Pye Limit Theorem does not hold for S2a-reducibility.