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.