Diophantine approximations with Fibonacci numbers
Journal de Théorie des Nombres de Bordeaux, Tome 25 (2013) no. 2, pp. 499-520.

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.

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.

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     title = {Diophantine approximations with {Fibonacci} numbers},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
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Victoria Zhuravleva. Diophantine approximations with Fibonacci numbers. Journal de Théorie des Nombres de Bordeaux, Tome 25 (2013) no. 2, pp. 499-520. doi : 10.5802/jtnb.846. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.846/

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