Diophantine approximations with Fibonacci numbers

TL;DR

This paper investigates how well real numbers can be approximated using Fibonacci numbers and related sequences, establishing optimal bounds for these approximations and their limits.

Contribution

It proves the best possible bounds for the approximation of real numbers by Fibonacci numbers and powers of the golden ratio, advancing understanding in Diophantine approximation.

Findings

Bound on the infimum of fractional parts involving Fibonacci numbers

Limit inferior bounds for Fibonacci-based approximations

Optimality of the established bounds

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.