Diophantine approximations with Fibonacci numbers
Journal de théorie des nombres de Bordeaux, Volume 25 (2013) no. 2, p. 499-520

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.

Soit ${F}_{n}$ le $n$-ième nombre de Fibonacci. Notons $\varphi =\frac{1+\sqrt{5}}{2}$. Nous prouvons les inégalités suivantes pour tous les nombres réels $\alpha$ :

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) ${lim inf}_{n\to \infty }||{\varphi }^{n}\alpha ||\le \frac{1}{5}$.

Ces résultats sont les meilleurs possibles.

@article{JTNB_2013__25_2_499_0,
author = {Zhuravleva, Victoria},
title = {Diophantine approximations with Fibonacci numbers},
journal = {Journal de th\'eorie des nombres de Bordeaux},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {25},
number = {2},
year = {2013},
pages = {499-520},
doi = {10.5802/jtnb.846},
mrnumber = {3228318},
zbl = {1283.11102},
language = {en},
url = {http://www.numdam.org/item/JTNB_2013__25_2_499_0}
}

Zhuravleva, Victoria. Diophantine approximations with Fibonacci numbers. Journal de théorie des nombres de Bordeaux, Volume 25 (2013) no. 2, pp. 499-520. doi : 10.5802/jtnb.846. http://www.numdam.org/item/JTNB_2013__25_2_499_0/

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