Extending the Limit Theorem of Barmpalias and Lewis-Pye to all reals

TL;DR

This paper generalizes a known limit theorem about Martin-Löf random reals and their approximations, extending it from left-c.e. reals to all reals using nondecreasing translation functions.

Contribution

It extends the limit theorem of Barmpalias and Lewis-Pye from left-c.e. reals to all reals by using nondecreasing translation functions.

Findings

The limit of the ratio of differences exists for all reals.

The result is equivalent to a formulation involving nondecreasing translation functions.

The theorem applies beyond left-c.e. reals to the entire set of reals.

Abstract

By a celebrated result of Ku\v{c}era and Slaman (DOI:10.1137/S0097539799357441), the Martin-L\"of random left-c.e. reals form the highest left-c.e. Solovay degree. Barmpalias and Lewis-Pye (arXiv:1604.00216) strengthened this result by showing that, for all left-c.e. reals $\alpha$ and $\beta$ such that $\beta$ is Martin-L\"of random and all left-c.e. approximations $a_0,a_1,\dots$ and $b_0,b_1,\dots$ of $\alpha$ and $\beta$, respectively, the limit \begin{equation*} \lim\limits_{n\to\infty}\frac{\alpha - a_n}{\beta - b_n} \end{equation*} exists and does not depend on the choice of the left-c.e. approximations to $\alpha$ and $\beta$. Here we give an equivalent formulation of the result of Barmpalias and Lewis-Pye in terms of nondecreasing translation functions and generalize their result to the set of all (i.e., not necessarily left-c.e.) reals.