The Fibonacci numbers are not a Heilbronn set

For a real number $\theta$, let $\mathrm{\parallel }\theta \mathrm{\parallel }$ denote the distance from $\theta$ to the nearest integer. A set of positive integers $\mathcal{H}$ is a Heilbronn set if for every $\alpha \in \mathbb{R}$ and every $\epsilon >0$ there exists $h\in \mathcal{H}$ such that $\mathrm{\parallel }h\alpha \mathrm{\parallel }<\epsilon$ (see \cite{montgomery} 2.7). The natural numbers are a Heilbronn set by Dirichlet’s approximation theorem. Vinogradov \cite{vinogradov} showed that for a natural number $k$, the $k$th powers of integers are a Heilbronn set. In this paper we give a constructive proof that the Fibonacci sequence is not a Heilbronn set, but conversely that almost all $\alpha$ satisfy ${\mathrm{lim\hspace{0.17em}inf}}_{n\to \mathrm{\infty }}\mathrm{\parallel }{F}_n\alpha \mathrm{\parallel }=0$. However, we exhibit a real number $\alpha$ such that $\mathrm{\parallel }{F}_n\alpha \mathrm{\parallel }>0.14$ for all $n$.