The Duffin-Schaeffer type conjectures in various local fields

Abstract: This paper discovers a new phenomenon about the Duffin-Schaeffer conjecture, which claims that $\lambda(\cap_{m=1}^{\infty}\cup_{n=m}^{\infty}{\mathcal E}n)=1$ if and only if $\sum_n\lambda({\mathcal E}_n)=\infty$, where $\lambda$ denotes the Lebesgue measure on $\mathbb{R}/\mathbb{Z}$, [ {\mathcal E}_n={\mathcal E}_n(\psi)=\bigcup{m=1 \atop (m,n)=1}^n\big(\frac{m-\psi(n)}{n},\frac{m+\psi(n)}{n}\big), ] $\psi$ 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 [\lambda(\bigcup{n=1}^{\infty}{\mathcal E}n)\geq C\min{\sum{n=1}^{\infty}\lambda({\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 $\lambda(\cap{m=1}^{\infty}\cup_{n=m}^{\infty}{\mathcal E}_n)=1$ 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.