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arXiv 1401.0035math.NT
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各种局部域中的Duffin-Schaeffer型结构

The Duffin-Schaeffer type conjectures in various local fields

Liangpan Li

发表信息：
Mathematika 62 (2016) 753-800
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摘要：

本文发现了Duffin-Schaeffer猜想中的一个新现象，即$λ(\cap_{m=1}^{\infty}\cup_{n=m}^{\infty}{\mathcal E}_n)=1$当且仅当$\sum_nλ({\mathcal E}_n)=\infty$，其中$λ$表示$\mathbb{R}/\mathbb{Z}$上的Lebesgue测度，\[ {\mathcal E}_n={\mathcal E}_n(ψ)=\bigcup_{m=1 \atop (m,n)=1}^n\big(\frac{m-ψ(n)}{n},\frac{m+ψ(n)}{n}\big), \] $ψ$是任意非负数论函数.本文不研究$\cap_{m=1}^{\infty}\cup_{n=m}^{\infty}{\mathcal E}_n$，而是引入了一个更基本的对象$\cup_{n=1}^{\infty}{\mathcal E}_n$，并猜想存在一个普适常数$C>0$，使得\[λ(\bigcup_{n=1}^{\infty}{\mathcal E}_n)\geq C\min\{\sum_{n=1}^{\infty}λ({\mathcal E}_n),1\}.\].证明了这个猜想等价于Duffin-Schaeffer猜想.在$p$-adic数和形式Laurent级数领域也发现了类似的现象.作为一个副产品，我们有条件地回答了海恩斯的问题，表明只要Duffin-Schaeffer猜想是真的，人们总是可以使用准独立的平均方法来推导$λ(\cap_{m=1}^{\infty}\cup_{n=m}^{\infty}{\mathcal E}_n)=1$。我们还表明，在其他几个人中，海恩斯，Pollington和Velani的两个代数等价于Duffin-Schaeffer猜想，并首次引入第二Borel-Cantelli引理的加权版本的Duffin-Schaeffer猜想的研究。

英文摘要：

This paper discovers a new phenomenon about the Duffin-Schaeffer conjecture, which claims that $λ(\cap_{m=1}^{\infty}\cup_{n=m}^{\infty}{\mathcal E}_n)=1$ if and only if $\sum_nλ({\mathcal E}_n)=\infty$, where $λ$ denotes the Lebesgue measure on $\mathbb{R}/\mathbb{Z}$, \[ {\mathcal E}_n={\mathcal E}_n(ψ)=\bigcup_{m=1 \atop (m,n)=1}^n\big(\frac{m-ψ(n)}{n},\frac{m+ψ(n)}{n}\big), \] $ψ$ is any non-negative arithmetical function. Instead of studying $\cap_{m=1}^{\infty}\cup_{n=m}^{\infty}{\mathcal E}_n$ we introduce an even fundamental object $\cup_{n=1}^{\infty}{\mathcal E}_n$ and conjecture there exists a universal constant $C>0$ such that \[λ(\bigcup_{n=1}^{\infty}{\mathcal E}_n)\geq C\min\{\sum_{n=1}^{\infty}λ({\mathcal E}_n),1\}.\] It is shown that this conjecture is equivalent to the Duffin-Schaeffer conjecture. Similar phenomena are found in the fields of $p$-adic numbers and formal Laurent series. As a byproduct, we answer conditionally a question of Haynes by showing that one can always use the quasi-independence on average method to deduce $λ(\cap_{m=1}^{\infty}\cup_{n=m}^{\infty}{\mathcal E}_n)=1$ as long as the Duffin-Schaeffer conjecture is true. We also show among several others that two conjectures of Haynes, Pollington and Velani are equivalent to the Duffin-Schaeffer conjecture, and introduce for the first time a weighted version of the second Borel-Cantelli lemma to the study of the Duffin-Schaeffer conjecture.

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