REGAININGLY APPROXIMABLE NUMBERS AND SETS.
We call an $\alpha \in \mathbb {R}$ regainingly approximable if there exists a computable nondecreasing sequence $(a_n)_n$ of rational numbers converging to $\alpha $ with $\alpha - a_n for infinitely many ${n \in \mathbb {N}}$. We also call a set $A\subseteq \mathbb {N}$ regainingly approximable if...
| Publicado en: | Journal of Symbolic Logic Vol. 90; no. 4; pp. 1664 - 1695 |
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| Autores principales: | , , |
| Formato: | Artículo |
| Publicado: |
Cambridge University Press
Dec2025
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| Materias: | |
| Acceso en línea: | Ver este registro en EBSCOhost |
| fields | @attributes: recordID: 1 pdfLink: plink: https://search.ebscohost.com/login.aspx?direct=true&db=hlh&AN=191386361&site=ehost-live header: @attributes: shortDbName: hlh uiTerm: 191386361 longDbName: Humanities International Complete uiTag: AN controlInfo: bkinfo: jinfo: jid: 00224812 3TY jtl: Journal of Symbolic Logic issn: 00224812 maglogo: N pubinfo: dt: Dec2025 vid: 90 iid: 4 pid: 15979 pub: Cambridge University Press artinfo: ui: 191386361 10.1017/jsl.2024.5 ppf: 1664 ppct: 31 formats: tig: atl: REGAININGLY APPROXIMABLE NUMBERS AND SETS. aug: au: HERTLING, PETER HÖLZL, RUPERT JANICKI, PHILIP affil: FAKULTÄT FÜR INFORMATIK UNIVERSITÄT DER BUNDESWEHR MÜNCHEN 85577 NEUBIBERG, GERMANY E-mail su: Recursive sequences (Mathematics) Real numbers Algorithmic randomness Recursion theory Kolmogorov complexity sug: subj: Recursive sequences (Mathematics) Real numbers Algorithmic randomness Recursion theory Kolmogorov complexity keyword: computably enumerable sets effective approximation left-computable numbers splitting Turing degrees ab: We call an $\alpha \in \mathbb {R}$ regainingly approximable if there exists a computable nondecreasing sequence $(a_n)_n$ of rational numbers converging to $\alpha $ with $\alpha - a_n for infinitely many ${n \in \mathbb {N}}$. We also call a set $A\subseteq \mathbb {N}$ regainingly approximable if it is c.e. and the strongly left-computable number $2^{-A}$ is regainingly approximable. We show that the set of regainingly approximable sets is neither closed under union nor intersection and that every c.e. Turing degree contains such a set. Furthermore, the regainingly approximable numbers lie properly between the computable and the left-computable numbers and are not closed under addition. While regainingly approximable numbers are easily seen to be i.o. K -trivial, we construct such an $\alpha $ such that ${K(\alpha \restriction n)>n}$ for infinitely many n. Similarly, there exist regainingly approximable sets whose initial segment complexity infinitely often reaches the maximum possible for c.e. sets. Finally, there is a uniform algorithm splitting regular real numbers into two regainingly approximable numbers that are still regular. pubtype: Academic Journal doctype: Article src: R language: English refInfo: copyright: @attributes: flag: Y dt: @attributes: year: 2025 holdings: @attributes: islocal: N |
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