Extending the Limit Theorem of Barmpalias and Lewis-Pye to all reals
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913437982916608 |
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| author | Titov, Ivan |
| author_facet | Titov, Ivan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_14445 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Extending the Limit Theorem of Barmpalias and Lewis-Pye to all reals Titov, Ivan Logic Information Theory Probability 03D32 (Primary) 68Q30, 03D30 (Secondary) 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. |
| title | Extending the Limit Theorem of Barmpalias and Lewis-Pye to all reals |
| topic | Logic Information Theory Probability 03D32 (Primary) 68Q30, 03D30 (Secondary) |
| url | https://arxiv.org/abs/2407.14445 |