The Fibonacci numbers are not a Heilbronn set

Abstract

For a real number $\theta$, let $\Vert\theta\Vert$ 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 $\Vert h\alpha\Vert<\epsilon$ (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 $\liminf_{n\to\infty}\Vert F_n\alpha\Vert=0$. However, we exhibit a real number $\alpha$ such that $\Vert F_n\alpha\Vert>0.14$ for all $n$.