The Duffin-Schaeffer conjecture with a moving target
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2024
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| author | Hauke, Manuel Ramirez, Felipe A. |
| author_facet | Hauke, Manuel Ramirez, Felipe A. |
| contents | We prove the inhomogeneous generalization of the Duffin-Schaeffer conjecture in dimension $m \geq 3$. That is, given $\mathbf{y}\in \mathbb{R}^m$ and $ψ:\mathbb{N}\to\mathbb{R}_{\geq 0}$ such that $\sum (φ(q)ψ(q)/q)^m = \infty$, we show that for almost every $\mathbf{x} \in\mathbb{R}^m$ there are infinitely many rational vectors $\mathbf{a}/q$ such that $\vert q\mathbf{x} - \mathbf{a} - \mathbf{y}\vert<ψ(q)$ and such that each component of $\mathbf{a}$ is coprime to $q$. This is an inhomogeneous extension of a homogeneous conjecture of Sprindžuk which was itself proved in 1990 by Pollington and Vaughan. In fact, our main result generalizes Pollington-Vaughan not only to the inhomogeneous case, but also to the setting of moving targets, where the inhomogeneous parameter $\mathbf{y}$ is free to vary with $q$. In contrast, we show by an explicit construction that the (1-dimensional) inhomogeneous Duffin-Schaeffer conjecture fails to hold with a moving target, implying that any successful attack on the one-dimensional problem must use the fact that the inhomogeneous parameter is constant. We also introduce new questions regarding moving targets. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_05344 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Duffin-Schaeffer conjecture with a moving target Hauke, Manuel Ramirez, Felipe A. Number Theory Dynamical Systems 11K60 11J83 11J71 11J20 11K50 We prove the inhomogeneous generalization of the Duffin-Schaeffer conjecture in dimension $m \geq 3$. That is, given $\mathbf{y}\in \mathbb{R}^m$ and $ψ:\mathbb{N}\to\mathbb{R}_{\geq 0}$ such that $\sum (φ(q)ψ(q)/q)^m = \infty$, we show that for almost every $\mathbf{x} \in\mathbb{R}^m$ there are infinitely many rational vectors $\mathbf{a}/q$ such that $\vert q\mathbf{x} - \mathbf{a} - \mathbf{y}\vert<ψ(q)$ and such that each component of $\mathbf{a}$ is coprime to $q$. This is an inhomogeneous extension of a homogeneous conjecture of Sprindžuk which was itself proved in 1990 by Pollington and Vaughan. In fact, our main result generalizes Pollington-Vaughan not only to the inhomogeneous case, but also to the setting of moving targets, where the inhomogeneous parameter $\mathbf{y}$ is free to vary with $q$. In contrast, we show by an explicit construction that the (1-dimensional) inhomogeneous Duffin-Schaeffer conjecture fails to hold with a moving target, implying that any successful attack on the one-dimensional problem must use the fact that the inhomogeneous parameter is constant. We also introduce new questions regarding moving targets. |
| title | The Duffin-Schaeffer conjecture with a moving target |
| topic | Number Theory Dynamical Systems 11K60 11J83 11J71 11J20 11K50 |
| url | https://arxiv.org/abs/2407.05344 |