Relative Randomness and Continuous Translation Functions

Abstract

The notions of Martin-Löf randomness and of Solovay reducibility, as well as their relations to each other, are central objects of study in algorithmic randomness. When restricted to left-c.e. reals, Solovay reducibility of α to β can be characterized [Cristian S. Calude et al., 2001] as the existence of two left approximations a₀,a₁,… and b₀,b₁,… of α and β, respectively, such that the ratios (α-a_n)/(β-b_n) are bounded from above. By a celebrated result of Kučera and Slaman, among left-c.e. reals the Martin-Löf random ones are largest with respect to Solovay reducibility. The latter result was largely improved by the Limit Theorem of Barmpalias and Lewis-Pye [George Barmpalias and Andrew Lewis-Pye, 2017], which asserts that for given left-c.e. reals α and β where β is Martin-Löf random, for all left-approximations of α and β as above, the ratios (α-a_n)/(β-b_n) converge to the same limit.
Though the original definition of Solovay reducibility applies to all reals, Solovay reducibility is considered to be badly behaved on the class of all reals. Accordingly, various variants of Solovay reducibility have been proposed, including variants defined via real-valued functions by Kumabe, Miyabe, and Suzuki [Kumabe et al., 2024] and a monotone variant by Titov [Titov, 2024]. It is known that for the monotone variant, the Limit Theorem of Barmpalias and Lewis-Pye extends to all reals [Titov, 2024]. By our main result, similarly the Limit Theorem holds for all reals with respect to the reducibility cl-open introduced by Kumabe et al. [Kumabe et al., 2024] in 2024. The result is formulated in terms of translation functions of bounded variation, and asserts that every such function from a Martin-Löf random real β to a real α is left differentiable in β. In a setting of functions that are required to be defined on the whole unit interval and not just on the reals strictly smaller than β, the differentiability of computable functions of bounded variation in every Martin-Löf random real was shown by Demuth [Demuth, 1975] in 1975; similar results for other types of computable functions and randomness notions were obtained by Brattka, Miller, and Nies [Vasco Brattka et al., 2011] in 2011 and Rute [Jason Rute, 2018] in 2018. 
Furthermore, we deduce from the main result an equivalent characterization of Martin-Löf randomness on the set of left-c.e. reals in terms of cl-open-reducibility of a real to itself.