Diophantine approximations with Fibonacci numbers

Details Diophantine approximations with Fibonacci numbers Diophantine approximations with Fibonacci numbers

2011-12-28

arxiv.org/abs/1112.6142v1

Mathematics

Number Theory

21 pages, 7 figures

Scientific paper

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.

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