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.


11J25 Diophantine inequalities
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
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