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THE DUFFIN–SCHAEFFER-TYPE CONJECTURES IN VARIOUS LOCAL FIELDS

Part of: Diophantine equations Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  06 May 2016

Liangpan Li
Liangpan Li*
Affiliation:
Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, U.K. email L.Li@lboro.ac.uk
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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.

MSC classification

Primary: 11J83: Metric theory
Secondary: 11D88: $p$-adic and power series fields
Type
Research Article
Information
Mathematika , Volume 62 , Issue 3 , 2016 , pp. 753 - 800
Copyright
Copyright © University College London 2016 

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