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The Ramanujan Journal

, Volume 46, Issue 3, pp 605–631 | Cite as

Continued fractions arising from \({\mathcal F}_{1,3}\)

  • S. Kushwaha
  • R. Sarma
Article
  • 88 Downloads

Abstract

We study a family of continued fractions arising from a graph known as \({\mathcal F}_{1,3}\) which is isomorphic to a subgraph of the Farey graph. We call these continued fractions \({\mathcal F}_{1,3}\)-continued fractions. In fact, certain paths from infinity to a vertex in \({\mathcal F}_{1,3}\) correspond to finite \({\mathcal F}_{1,3}\)-continued fractions of the vertex and vice versa. Further, we study uniqueness of the longest \({\mathcal F}_{1,3}\)-continued fraction expansions of real numbers and show that their convergents are best approximations of the numbers by vertices of \({\mathcal F}_{1,3}\).

Keywords

Continued fraction Convergents Best approximation Farey graph 

Mathematics Subject Classification

Primary 11A55 11J70 

Notes

Acknowledgements

The authors are grateful to the anonymous referee for his/her helpful comments.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology DelhiDelhiIndia

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