Research output: Contribution to journal › Article › Scientific › peer-review

**Natural extensions for Nakada's α-expansions : Descending from 1 to g ^{2}.** / de Jonge, Jaap; Kraaikamp, Cornelis.

Research output: Contribution to journal › Article › Scientific › peer-review

de Jonge, J & Kraaikamp, C 2018, 'Natural extensions for Nakada's α-expansions: Descending from 1 to g^{2}', *Journal of Number Theory*, vol. 183, pp. 172-212. https://doi.org/10.1016/j.jnt.2017.07.012

de Jonge, J., & Kraaikamp, C. (2018). Natural extensions for Nakada's α-expansions: Descending from 1 to g^{2}. *Journal of Number Theory*, *183*, 172-212. https://doi.org/10.1016/j.jnt.2017.07.012

de Jonge J, Kraaikamp C. Natural extensions for Nakada's α-expansions: Descending from 1 to g^{2}. Journal of Number Theory. 2018 Feb;183:172-212. https://doi.org/10.1016/j.jnt.2017.07.012

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title = "Natural extensions for Nakada's α-expansions: Descending from 1 to g2",

abstract = "By means of singularisations and insertions in Nakada's α-expansions, which involves the removal of partial quotients 1 while introducing partial quotients with a minus sign, the natural extension of Nakada's continued fraction map Tα is given for (10-2)/3≤α<1. From our construction it follows that Ωα, the domain of the natural extension of Tα, is metrically isomorphic to Ωg for α∈[g2,g), where g is the small golden mean. Finally, although Ωα proves to be very intricate and unmanageable for α∈[g2,(10-2)/3), the α-Legendre constant L(α) on this interval is explicitly given.",

keywords = "Continued fractions, Metric theory",

author = "{de Jonge}, Jaap and Cornelis Kraaikamp",

note = "Accepted Author Manuscript",

year = "2018",

month = feb,

doi = "10.1016/j.jnt.2017.07.012",

language = "English",

volume = "183",

pages = "172--212",

journal = "Journal of Number Theory",

issn = "0022-314X",

publisher = "Academic Press",

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T1 - Natural extensions for Nakada's α-expansions

T2 - Descending from 1 to g2

AU - de Jonge, Jaap

AU - Kraaikamp, Cornelis

N1 - Accepted Author Manuscript

PY - 2018/2

Y1 - 2018/2

N2 - By means of singularisations and insertions in Nakada's α-expansions, which involves the removal of partial quotients 1 while introducing partial quotients with a minus sign, the natural extension of Nakada's continued fraction map Tα is given for (10-2)/3≤α<1. From our construction it follows that Ωα, the domain of the natural extension of Tα, is metrically isomorphic to Ωg for α∈[g2,g), where g is the small golden mean. Finally, although Ωα proves to be very intricate and unmanageable for α∈[g2,(10-2)/3), the α-Legendre constant L(α) on this interval is explicitly given.

AB - By means of singularisations and insertions in Nakada's α-expansions, which involves the removal of partial quotients 1 while introducing partial quotients with a minus sign, the natural extension of Nakada's continued fraction map Tα is given for (10-2)/3≤α<1. From our construction it follows that Ωα, the domain of the natural extension of Tα, is metrically isomorphic to Ωg for α∈[g2,g), where g is the small golden mean. Finally, although Ωα proves to be very intricate and unmanageable for α∈[g2,(10-2)/3), the α-Legendre constant L(α) on this interval is explicitly given.

KW - Continued fractions

KW - Metric theory

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