The Duffin–Schaeffer theorem in number fields
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 (D. G. Cantor, 1965), and a version of the Duffin–Schaeffer theorem proven only in imaginary quadratic fields (H. Nakada and G. Wagner, 1991).
In this paper, we prove a version of the Duffin–Schaeffer theorem for all number fields.