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THE DUFFIN–SCHAEFFER-TYPE CONJECTURES IN VARIOUS LOCAL FIELDS

  • Liangpan Li (a1)

Abstract

In this paper we study the Duffin–Schaeffer conjecture, which claims that $\unicode[STIX]{x1D706}(\bigcap _{m=1}^{\infty }\bigcup _{n=m}^{\infty }{\mathcal{E}}_{n})=1$ if and only if $\sum _{n=1}^{\infty }\unicode[STIX]{x1D706}({\mathcal{E}}_{n})=\infty$ , where $\unicode[STIX]{x1D706}$ denotes the Lebesgue measure on $\mathbb{R}/\mathbb{Z}$ ,

$$\begin{eqnarray}{\mathcal{E}}_{n}={\mathcal{E}}_{n}(\unicode[STIX]{x1D713})=\mathop{\bigcup }_{\substack{ m=1 \\ (m,n)=1}}^{n}\bigg(\frac{m-\unicode[STIX]{x1D713}(n)}{n},\frac{m+\unicode[STIX]{x1D713}(n)}{n}\bigg),\end{eqnarray}$$
and $\unicode[STIX]{x1D713}$ denotes any non-negative arithmetical function. Instead of studying the superior limit $\bigcap _{m=1}^{\infty }\bigcup _{n=m}^{\infty }{\mathcal{E}}_{n}$ we focus on the union $\bigcup _{n=1}^{\infty }{\mathcal{E}}_{n}$ and conjecture that there exists a universal constant $C>0$ such that
$$\begin{eqnarray}\unicode[STIX]{x1D706}\bigg(\mathop{\bigcup }_{n=1}^{\infty }{\mathcal{E}}_{n}\bigg)\geqslant C\min \bigg\{\mathop{\sum }_{n=1}^{\infty }\unicode[STIX]{x1D706}({\mathcal{E}}_{n}),1\bigg\}.\end{eqnarray}$$
It is shown that this conjecture is equivalent to the Duffin–Schaeffer conjecture. Similar phenomena exist in the fields of $p$ -adic numbers and formal Laurent series. Furthermore, two conjectures of Haynes, Pollington and Velani are shown to be equivalent to the Duffin–Schaeffer conjecture, and a weighted version of the second Borel–Cantelli lemma is introduced to study the Duffin–Schaeffer conjecture.

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