Vaaler's theorem in number fields

TL;DR

This paper extends Vaaler's theorem, a key result in Diophantine approximation, from the rational numbers and imaginary quadratic fields to general number fields, advancing understanding in this area.

Contribution

It generalizes Vaaler's theorem to all number fields, broadening its applicability in Diophantine approximation and number theory.

Findings

Vaaler's theorem is established for general number fields.

The result connects to the Duffin--Schaeffer conjecture.

Advances the theoretical framework in Diophantine approximation.

Abstract

In Diophantine approximation, Vaaler's theorem was an important partial result towards the Duffin--Schaeffer conjecture, which was open for almost eighty years before it was recently proven by Koukoulopoulos and Maynard. A version of this result was previously proven to also hold in imaginary quadratic fields: in this paper, we establish a version of Vaaler's theorem in general number fields.