Diophantine approximations with Fibonacci numbers
  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. 
Victoria Zhuravleva
math.NT