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

By a celebrated result of Kučera and Slaman (DOI:10.1137/S0097539799357441), the Martin-Löf 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 $α$ and $β$ such that $β$ is Martin-Löf random and all left-c.e. approximations $a_0,a_1,\dots$ and $b_0,b_1,\dots$ of $α$ and $β$, respectively, the limit \begin{equation*} \lim\limits_{n\to\infty}\frac{α- a_n}{β- b_n} \end{equation*} exists and does not depend on the choice of the left-c.e. approximations to $α$ and $β$. 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.