Refinement of the Theorem of Vahlen

Bhattarai, Dinesh Sharma

doi:10.2478/udt-2023-0001

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Refinement of the Theorem of Vahlen

Uniform distribution theory

Volume 18 (2023): Issue 1 (July 2023)

By: Dinesh Sharma Bhattarai

Open Access

|Aug 2023

Abstract

In 1895, Vahlen proved a theorem concerning a simultaneous approximation of a real number by its two consecutive convergent. In this paper, we will provide a sharper bound for the theorem.

References

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HANČL, J.—BAHNEROVÁ, S.: Sharpening of the theorem of Vahlen and related theorems, J. Ramanujan Math. Soc. 36 (2021), no. 2, 109–121.
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KHINCHIN, A. I.—KHINTCHINE, A. YA.: Continued Fractions Vol. 525.Translated by Peter P. Wynn, Noordhoff Ltd., Groningen 1963.
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SCHMIDT, W. M.: Diophantine Approximation.In: Lectur Notes in Math. Vol. 785. Springer Science and Business Media, 1996.
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VAHLEN, K. TH.: Ueber Näherungswerte und Kettenbrüche,J. Reine Angew. Math. 115 (1895), 221–233. https://doi.org/10.1515/crll.1895.115.221
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DOI: https://doi.org/10.2478/udt-2023-0001 | Journal eISSN: 2309-5377 | Journal ISSN: 1336-913X

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Language: English

Page range: 1 - 8

Submitted on: Jun 13, 2022

Accepted on: Dec 18, 2022

Published on: Aug 10, 2023

Published by: Slovak Academy of Sciences, Mathematical Institute

In partnership with: Paradigm Publishing Services

Publication frequency: 2 issues per year

Keywords:

continued fractions,

diophantine approximation,

consecutive convergents

Related subjects:

Mathematics,

Algebra and number theory,

Discrete mathematics,

Probability and statistics,

Numerical and computational mathematics

© 2023 Dinesh Sharma Bhattarai, published by Slovak Academy of Sciences, Mathematical Institute
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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