The Duffin-Schaeffer theorem in number fields

TL;DR

This paper extends the Duffin-Schaeffer theorem, a key result in metric number theory, to all number fields, broadening its applicability beyond previously known cases.

Contribution

It provides the first proof of the Duffin-Schaeffer theorem applicable to all number fields, generalizing prior results limited to imaginary quadratic fields.

Findings

Proves the Duffin-Schaeffer theorem for all number fields.

Establishes a unified framework for Diophantine approximation in number fields.

Enhances understanding of metric number theory in algebraic number settings.

Abstract

The Duffin-Schaeffer theorem is a well-known result from metric number theory, which generalises Khinchin's theorem from monotonic functions to a wider class of approximating functions. In recent years, there has been some interest in proving versions of classical theorems from Diophantine approximation in various generalised settings. In the case of number fields, there has been a version of Khinchin's theorem proven which holds for all number fields, and a version of the Duffin-Schaeffer theorem proven only in imaginary quadratic fields. In this paper, we prove a version of the Duffin-Schaeffer theorem for all number fields.