Erdős's integer dilation approximation problem and GCD graphs
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912231498711040 |
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| author | Koukoulopoulos, Dimitris Lamzouri, Youness Lichtman, Jared Duker |
| author_facet | Koukoulopoulos, Dimitris Lamzouri, Youness Lichtman, Jared Duker |
| contents | Let $\mathcal{A}\subset\mathbb{R}_{\geqslant1}$ be a countable set such that $\limsup_{x\to\infty}\frac{1}{\log x}\sum_{α\in\mathcal{A}\cap[1,x]}\frac{1}α>0$. We prove that, for every $\varepsilon>0$, there exist infinitely many pairs $(α, β)\in \mathcal{A}^2$ such that $α\neq β$ and $|nα-β| <\varepsilon$ for some positive integer $n$. This resolves a problem of Erdős from 1948. A critical role in the proof is played by the machinery of GCD graphs, which were introduced by the first author and by James Maynard in their work on the Duffin--Schaeffer conjecture in Diophantine approximation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_09539 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Erdős's integer dilation approximation problem and GCD graphs Koukoulopoulos, Dimitris Lamzouri, Youness Lichtman, Jared Duker Number Theory Combinatorics Dynamical Systems Let $\mathcal{A}\subset\mathbb{R}_{\geqslant1}$ be a countable set such that $\limsup_{x\to\infty}\frac{1}{\log x}\sum_{α\in\mathcal{A}\cap[1,x]}\frac{1}α>0$. We prove that, for every $\varepsilon>0$, there exist infinitely many pairs $(α, β)\in \mathcal{A}^2$ such that $α\neq β$ and $|nα-β| <\varepsilon$ for some positive integer $n$. This resolves a problem of Erdős from 1948. A critical role in the proof is played by the machinery of GCD graphs, which were introduced by the first author and by James Maynard in their work on the Duffin--Schaeffer conjecture in Diophantine approximation. |
| title | Erdős's integer dilation approximation problem and GCD graphs |
| topic | Number Theory Combinatorics Dynamical Systems |
| url | https://arxiv.org/abs/2502.09539 |