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 ϕ=1+5 2. We prove that the following inequalities hold for any real α:

1) inf 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 ||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.

@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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