EXTENDING THE BARMPALIAS–LEWIS-PYE LIMIT THEOREM TO ALL REALS

Abstract By a celebrated result of Kučera and Slaman [5], the Martin-Löf random left-c.e. reals form the highest left-c.e. Solovay degree. Barmpalias and Lewis-Pye [1] strengthened this result by showing that, for all left-c.e. reals $\alpha $ and $\beta $ such that $\beta $ is Martin-Löf 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 $$ \begin{align*} \lim\limits_{n\to\infty}\frac{\alpha - a_n}{\beta - b_n} \end{align*} $$ 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.