Diophantine approximations with Fibonacci numbers

@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 -