The Duffin-Schaeffer type conjectures in various local fields

This paper discovers a new phenomenon about the Duffin-Schaeffer conjecture, which claims that $\lambda(\cap_{m=1}^{\infty}\cup_{n=m}^{\infty}{\mathcal E}_n)=1$ifandonlyif$\sum_n\lambda({\mathcal E}_n)=\infty$,where$\lambda$denotestheLebesguemeasureon$$\mathbb{R}$/$\mathbb{Z}$$, $${\mathcal{E}}_n={\mathcal{E}}_n(\psi )=\bigcup _{\genfrac{}{}{0px}{}{m=1}{(m,n)=1}}^n(\frac{m-\psi (n)}{n},\frac{m+\psi (n)}{n}),$$ $\psi$ is any non-negative arithmetical function. Instead of studying ${\cap }_{m=1}^{\mathrm{\infty }}{\cup }_{n=m}^{\mathrm{\infty }}{\mathcal{E}}_n$ we introduce an even fundamental object ${\cup }_{n=1}^{\mathrm{\infty }}{\mathcal{E}}_n$ and conjecture there exists a universal constant $C>0$ such that $$\lambda (\bigcup _{n=1}^{\mathrm{\infty }}{\mathcal{E}}_n)\ge C\min \{\sum _{n=1}^{\mathrm{\infty }}\lambda ({\mathcal{E}}_n),1\}\mathrm{.}$$ 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 $\lambda ({\cap }_{m=1}^{\mathrm{\infty }}{\cup }_{n=m}^{\mathrm{\infty }}{\mathcal{E}}_n)=1$ 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.