The Duffin-Schaeffer Conjecture for multiplicative Diophantine approximation
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arXiv
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| Format: | Preprint |
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2024
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| author | Frühwirth, Lorenz Hauke, Manuel |
| author_facet | Frühwirth, Lorenz Hauke, Manuel |
| contents | Given a monotonically decreasing $ψ: \mathbb{N} \to [0,\infty)$, Khintchine's Theorem provides an efficient tool to decide whether, for almost every $α\in \mathbb{R}$, there are infinitely many $(p,q) \in \mathbb{Z}^2$ such that $\left\lvert α- \frac{p}{q}\right\rvert \leq \frac{ψ(q)}{q}$. The recent result of Koukoulopoulos and Maynard provides an elegant way of removing monotonicity when only counting reduced fractions. Gallagher showed a multiplicative higher-dimensional generalization to Khintchine's Theorem, again assuming monotonicity. In this article, we prove the following Duffin-Schaeffer-type result for multiplicative approximations: For any $k\geq 1$, any function $ψ: \mathbb{N} \to [0,1/2]$ (not necessarily monotonic) and almost every $α\in \mathbb{R}^k$, there exist infinitely many $q$ such that $\prod\limits_{i=1}^k \left\lvert α_i - \frac{p_i}{q}\right\rvert \leq \frac{ψ(q)}{q^k}, p_1,\ldots,p_k$ all coprime to $q$, if and only if \[\sum\limits_{q \in \mathbb{N}} ψ(q) \left(\frac{φ(q)}{q} \right)^k\log \left(\frac{q}{φ(q)ψ(q)}\right)^{k-1} = \infty.\] This settles a conjecture of Beresnevich, Haynes, and Velani. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_11257 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Duffin-Schaeffer Conjecture for multiplicative Diophantine approximation Frühwirth, Lorenz Hauke, Manuel Number Theory 11K60 (Primary) 11J83, 11J71, 11H16 (Secondary) Given a monotonically decreasing $ψ: \mathbb{N} \to [0,\infty)$, Khintchine's Theorem provides an efficient tool to decide whether, for almost every $α\in \mathbb{R}$, there are infinitely many $(p,q) \in \mathbb{Z}^2$ such that $\left\lvert α- \frac{p}{q}\right\rvert \leq \frac{ψ(q)}{q}$. The recent result of Koukoulopoulos and Maynard provides an elegant way of removing monotonicity when only counting reduced fractions. Gallagher showed a multiplicative higher-dimensional generalization to Khintchine's Theorem, again assuming monotonicity. In this article, we prove the following Duffin-Schaeffer-type result for multiplicative approximations: For any $k\geq 1$, any function $ψ: \mathbb{N} \to [0,1/2]$ (not necessarily monotonic) and almost every $α\in \mathbb{R}^k$, there exist infinitely many $q$ such that $\prod\limits_{i=1}^k \left\lvert α_i - \frac{p_i}{q}\right\rvert \leq \frac{ψ(q)}{q^k}, p_1,\ldots,p_k$ all coprime to $q$, if and only if \[\sum\limits_{q \in \mathbb{N}} ψ(q) \left(\frac{φ(q)}{q} \right)^k\log \left(\frac{q}{φ(q)ψ(q)}\right)^{k-1} = \infty.\] This settles a conjecture of Beresnevich, Haynes, and Velani. |
| title | The Duffin-Schaeffer Conjecture for multiplicative Diophantine approximation |
| topic | Number Theory 11K60 (Primary) 11J83, 11J71, 11H16 (Secondary) |
| url | https://arxiv.org/abs/2403.11257 |