Diophantine approximations with Fibonacci numbers

Abstract: Let $F_{n}$ be the $n$-th Fibonacci number. Put $\varphi=\frac{1+\sqrt5}{2}$. We prove that the following inequalities hold for any real $\alpha$: 1) $\inf_{n \in \mathbb N} ||F_n\alpha||\le\frac{\varphi-1}{\varphi+2}$, 2) $\liminf_{n\to \infty}||F_n\alpha||\le 1/5$, 3) $\liminf_{n \to \infty}||\varphi^n \alpha||\le 1/5$. These results are the best possible.