Bounded tiles in $\mathbb{Q}_p$ are compact open sets

TL;DR

This paper proves that any bounded tile in the p-adic number field  is essentially a compact open set, providing a straightforward proof of this property.

Contribution

The paper offers a simple, direct proof that bounded tiles in  are compact open sets up to measure-zero differences.

Findings

Bounded tiles in  are compact open sets.

The proof simplifies understanding of tiling properties in p-adic fields.

Abstract

Any bounded tile of the field $\mathbb{Q}_p$ of $p$-adic numbers is a compact open set up to a zero Haar measure set. In this note, we give a simple and direct proof of this fact.