α-Expansions with odd partial quotients

Florin P. Boca, Claire Merriman

Research output: Contribution to journalArticlepeer-review

Abstract

We consider an analogue of Nakada's α-continued fraction transformation in the setting of continued fractions with odd partial quotients. More precisely, given α∈[[Formula presented](5−1),[Formula presented](5+1)], we show that every irrational number x∈I α =[α−2,α) can be uniquely represented as with e i (x;α)∈{±1} and d i (x;α)∈2N−1 determined by the iterates of the transformation φ α (x):=[Formula presented]−2[[Formula presented]+[Formula presented]]−1 of I α . We also describe the natural extension of φ α and prove that the endomorphism φ α is exact.

Original languageEnglish (US)
Pages (from-to)322-341
Number of pages20
JournalJournal of Number Theory
Volume199
DOIs
StatePublished - Jun 2019

Keywords

  • Ergodicity
  • Gauus map
  • Invariant measure
  • Natural extension
  • Odd continued fractions
  • α-Expansions

ASJC Scopus subject areas

  • Algebra and Number Theory

Fingerprint Dive into the research topics of 'α-Expansions with odd partial quotients'. Together they form a unique fingerprint.

Cite this