# Diophantine approximations with Fibonacci numbers

• [1] Moscow Lomonosov State University Department of Number Theory
• Volume: 25, Issue: 2, page 499-520
• ISSN: 1246-7405

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## Abstract

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Let ${F}_{n}$ be the $n$-th Fibonacci number. Put $\varphi =\frac{1+\sqrt{5}}{2}$. We prove that the following inequalities hold for any real $\alpha$:1) ${inf}_{n\in ℕ}||{F}_{n}\alpha ||\le \frac{\varphi -1}{\varphi +2}$,2) ${lim inf}_{n\to \infty }||{F}_{n}\alpha ||\le \frac{1}{5}$,3) ${lim inf}_{n\to \infty }||{\varphi }^{n}\alpha ||\le \frac{1}{5}$.These results are the best possible.

## How to cite

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Zhuravleva, Victoria. "Diophantine approximations with Fibonacci numbers." Journal de Théorie des Nombres de Bordeaux 25.2 (2013): 499-520. <http://eudml.org/doc/275708>.

@article{Zhuravleva2013,
abstract = {Let $F_\{n\}$ be the $n$-th Fibonacci number. Put $\varphi =\frac\{1+\sqrt\{5\}\}\{2\}$. We prove that the following inequalities hold for any real $\alpha$:1) $\inf _\{n \in \mathbb\{N\} \} ||F_n\alpha ||\le \frac\{\varphi -1\}\{\varphi +2\}$,2) $\liminf _\{n\rightarrow \infty \}||F_n\alpha ||\le \frac\{1\}\{5\}$,3) $\liminf _\{n \rightarrow \infty \}||\varphi ^n \alpha ||\le \frac\{1\}\{5\}$.These results are the best possible.},
affiliation = {Moscow Lomonosov State University Department of Number Theory},
author = {Zhuravleva, Victoria},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {diophantine approximation; Fibonacci number; algebraic number},
language = {eng},
month = {9},
number = {2},
pages = {499-520},
publisher = {Société Arithmétique de Bordeaux},
title = {Diophantine approximations with Fibonacci numbers},
url = {http://eudml.org/doc/275708},
volume = {25},
year = {2013},
}

TY - JOUR
AU - Zhuravleva, Victoria
TI - Diophantine approximations with Fibonacci numbers
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2013/9//
PB - Société Arithmétique de Bordeaux
VL - 25
IS - 2
SP - 499
EP - 520
AB - Let $F_{n}$ be the $n$-th Fibonacci number. Put $\varphi =\frac{1+\sqrt{5}}{2}$. We prove that the following inequalities hold for any real $\alpha$:1) $\inf _{n \in \mathbb{N} } ||F_n\alpha ||\le \frac{\varphi -1}{\varphi +2}$,2) $\liminf _{n\rightarrow \infty }||F_n\alpha ||\le \frac{1}{5}$,3) $\liminf _{n \rightarrow \infty }||\varphi ^n \alpha ||\le \frac{1}{5}$.These results are the best possible.
LA - eng
KW - diophantine approximation; Fibonacci number; algebraic number
UR - http://eudml.org/doc/275708
ER -

## References

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1. R. K. Akhunzhanov, On the distribution modulo 1 of exponential sequences. Mathematical notes 76:2 (2004), 153–160. Zbl1196.11107MR2098988
2. A. Dubickas, Arithmetical properties of powers of algebraic numbers. Bull. London Math. Soc. 38 (2006), 70–80. Zbl1164.11025MR2201605
3. L. Kuipers, H. Niederreiter, Uniform distribution of sequences. John Wiley & Sons, 1974. Zbl0281.10001MR419394
4. W. M. Schmidt, Diophantine approximations. Lect. Not. Math. 785, 1980. Zbl0421.10019
5. W. M. Schmidt, On badly approximable numbers and certain games. Trans. Amer. Math. Soc. 623 (1966), 178–199. Zbl0232.10029MR195595

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