## Diophantine approximations with Fibonacci numbers.(English. French summary)Zbl 1283.11102

Summary: Let $$F_n$$ be the $$n$$th Fibonacci number. Put $$\varphi =\frac{1+\sqrt 5}2$$. The author proves that the following inequalities hold for any real $$\alpha$$:
1) $$\inf_{n\in \mathbb N } ||F_n\alpha ||\leq \frac{\varphi-1}{\varphi+2}$$,
2) $$\liminf_{ n\to\infty }||F_n\alpha ||\leq \frac 15$$,
3) $$\liminf_{n\to\infty}||\varphi^n\alpha ||\leq \frac 15$$.
These bounds are sharp.

### MSC:

 11J25 Diophantine inequalities 11B39 Fibonacci and Lucas numbers and polynomials and generalizations
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### References:

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