Diophantine approximations with Fibonacci numbers
Victoria Zhuravleva1
1 Moscow Lomonosov State University Department of Number Theory
Journal de théorie des nombres de Bordeaux, Volume 25 (2013) no. 2, pp. 499-520.

Let F n be the n-th Fibonacci number. Put ϕ=1+5 2. We prove that the following inequalities hold for any real α:

1) inf n∈ℕ ||F n α||≤ϕ-1 ϕ+2,

2) lim inf n→∞ ||F n α||≤1 5,

3) lim inf n→∞ ||ϕ n α||≤1 5.

These results are the best possible.

Soit F n le n-ième nombre de Fibonacci. Notons ϕ=1+5 2. Nous prouvons les inégalités suivantes pour tous les nombres réels α :

1) inf n∈ℕ ||F n α||≤ϕ-1 ϕ+2,

2) lim inf n→∞ ||F n α||≤1 5,

3) lim inf n→∞ ||ϕ n α||≤1 5.

Ces résultats sont les meilleurs possibles.

DOI: 10.5802/jtnb.846

Victoria Zhuravleva 1

1 Moscow Lomonosov State University Department of Number Theory 
@article{JTNB_2013__25_2_499_0,
     author = {Victoria Zhuravleva},
     title = {Diophantine approximations with {Fibonacci} numbers},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {499--520},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {25},
     number = {2},
     year = {2013},
     doi = {10.5802/jtnb.846},
     mrnumber = {3228318},
     zbl = {1283.11102},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.846/}
}
TY  - JOUR
AU  - Victoria Zhuravleva
TI  - Diophantine approximations with Fibonacci numbers
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2013
SP  - 499
EP  - 520
VL  - 25
IS  - 2
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.846/
DO  - 10.5802/jtnb.846
LA  - en
ID  - JTNB_2013__25_2_499_0
ER  - 
%0 Journal Article
%A Victoria Zhuravleva
%T Diophantine approximations with Fibonacci numbers
%J Journal de théorie des nombres de Bordeaux
%D 2013
%P 499-520
%V 25
%N 2
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.846/
%R 10.5802/jtnb.846
%G en
%F JTNB_2013__25_2_499_0
Victoria Zhuravleva. 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. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.846/

[1] R. K. Akhunzhanov, On the distribution modulo 1 of exponential sequences. Mathematical notes 76:2 (2004), 153–160. | MR | Zbl

[2] A. Dubickas, Arithmetical properties of powers of algebraic numbers. Bull. London Math. Soc. 38 (2006), 70–80. | MR | Zbl

[3] L. Kuipers, H. Niederreiter, Uniform distribution of sequences. John Wiley & Sons, 1974. | MR | Zbl

[4] W. M. Schmidt, Diophantine approximations. Lect. Not. Math. 785, 1980. | Zbl

[5] W. M. Schmidt, On badly approximable numbers and certain games. Trans. Amer. Math. Soc. 623 (1966), 178–199. | MR | Zbl

Cited by Sources: