Natural extensions for Nakada's α-expansions: Descending from 1 to g2

Jaap de Jonge*, Cornelis Kraaikamp

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

2 Citations (Scopus)
37 Downloads (Pure)


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.

Original languageEnglish
Pages (from-to)172-212
Number of pages41
JournalJournal of Number Theory
Publication statusPublished - Feb 2018

Bibliographical note

Accepted Author Manuscript


  • Continued fractions
  • Metric theory


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