The $p$-adic Duffin--Schaeffer conjecture

TL;DR

This paper proves Haynes' version of the Duffin--Schaeffer conjecture in the context of $p$-adic numbers and explores related conjectures in $p$-adic approximation inspired by Jarník and Lutz.

Contribution

It establishes the $p$-adic Duffin--Schaeffer conjecture for Haynes' version and investigates related false conjectures in $p$-adic approximation.

Findings

Proof of Haynes' $p$-adic Duffin--Schaeffer conjecture

Results on related false conjectures in $p$-adic approximation

Insights into $p$-adic Diophantine approximation theories

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 Jarn\'ik and Lutz.