A unifying theory for metrical results on regular continued fraction convergents and mediants
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2023
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866911252128727040 |
|---|---|
| author | Dajani, Karma Kraaikamp, Cor Sanderson, Slade |
| author_facet | Dajani, Karma Kraaikamp, Cor Sanderson, Slade |
| contents | We revisit Ito's (\cite{I1989}) natural extension of the Farey tent map, which generates all regular continued fraction convergents and mediants of a given irrational. With a slight shift in perspective on the order in which these convergents and mediants arise, this natural extension is shown to provide an elegant and powerful tool in the metric theory of continued fractions. A wealth of old and new results -- including limiting distributions of approximation coefficients, analogues of a theorem of Legendre and their refinements, and a generalisation of Lévy's Theorem to subsequences of convergents and mediants -- are presented as corollaries within this unifying theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_13988 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A unifying theory for metrical results on regular continued fraction convergents and mediants Dajani, Karma Kraaikamp, Cor Sanderson, Slade Dynamical Systems Number Theory 11K50 (Primary) 11A55, 11J70 (Secondary) We revisit Ito's (\cite{I1989}) natural extension of the Farey tent map, which generates all regular continued fraction convergents and mediants of a given irrational. With a slight shift in perspective on the order in which these convergents and mediants arise, this natural extension is shown to provide an elegant and powerful tool in the metric theory of continued fractions. A wealth of old and new results -- including limiting distributions of approximation coefficients, analogues of a theorem of Legendre and their refinements, and a generalisation of Lévy's Theorem to subsequences of convergents and mediants -- are presented as corollaries within this unifying theory. |
| title | A unifying theory for metrical results on regular continued fraction convergents and mediants |
| topic | Dynamical Systems Number Theory 11K50 (Primary) 11A55, 11J70 (Secondary) |
| url | https://arxiv.org/abs/2312.13988 |