The $p$-adic Duffin--Schaeffer conjecture

Abstract

We prove Haynes' version of the Duffin--Schaeffer conjecture for the $p$-adic numbers. In addition, we prove several results about an associated related but false conjecture, related to $p$-adic approximation in the spirit of Jarník and Lutz.

Typos corrected and clarifications made, following suggestions of a referee