Attainable measures for certain types of p-adic Duffin--Schaeffer sets

TL;DR

This paper resolves recent conjectures about the $p$-adic Haar measure of certain Diophantine approximation sets by analyzing their measure spectrum, leading to a contradiction of the conjectures.

Contribution

It determines the measure spectrum of $p$-adic Duffin--Schaeffer sets, providing a definitive answer to longstanding conjectures in the field.

Findings

The measure spectrum of the sets is fully characterized.

The conjectures about the measure are disproved.

The results clarify the structure of $p$-adic Diophantine sets.

Abstract

This paper settles recent conjectures concerning the $p$-adic Haar measure applied to a family of sets defined in terms of Diophantine approximation. This is done by determining the spectrum of measure values for each family and seeing that this contradicts the corresponding conjectures.