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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2503.01788 |
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| _version_ | 1866916642193145856 |
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| author | Shiu, Daniel |
| author_facet | Shiu, Daniel |
| contents | For a real number $θ$, let $\Vertθ\Vert$ denote the distance from $θ$ to the nearest integer. A set of positive integers $\mathcal H$ is a Heilbronn set if for every $α\in \mathbb R$ and every $ε>0$ there exists $h\in\mathcal H$ such that $\Vert hα\Vert<ε$ (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 $α$ satisfy $\liminf_{n\to\infty}\Vert F_nα\Vert=0$. However, we exhibit a real number $α$ such that $\Vert F_nα\Vert>0.14$ for all $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_01788 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Fibonacci numbers are not a Heilbronn set Shiu, Daniel Number Theory 11B39, 11J71, 11K38 For a real number $θ$, let $\Vertθ\Vert$ denote the distance from $θ$ to the nearest integer. A set of positive integers $\mathcal H$ is a Heilbronn set if for every $α\in \mathbb R$ and every $ε>0$ there exists $h\in\mathcal H$ such that $\Vert hα\Vert<ε$ (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 $α$ satisfy $\liminf_{n\to\infty}\Vert F_nα\Vert=0$. However, we exhibit a real number $α$ such that $\Vert F_nα\Vert>0.14$ for all $n$. |
| title | The Fibonacci numbers are not a Heilbronn set |
| topic | Number Theory 11B39, 11J71, 11K38 |
| url | https://arxiv.org/abs/2503.01788 |