Tilings of the infinite $p$-ary tree and Cantor homeomorphisms

TL;DR

This paper introduces a comprehensive framework for tiling infinite p-ary trees, linking geometric, algebraic, and topological perspectives, and applies it to construct homeomorphisms between p-adic integer spaces.

Contribution

It establishes equivalent criteria for tiling infinite p-ary trees and uses these to explicitly construct homeomorphisms between different p-adic integer spaces.

Findings

Defined a notion of tiling for infinite p-ary trees

Established multiple equivalent criteria for tiles

Constructed explicit homeomorphisms between p-adic spaces

Abstract

We define a notion of tiling of the full infinite $p$-ary tree, establishing a series of equivalent criteria for a subtree to be a tile, each of a different nature; namely, geometric, algebraic, graph-theoretic, order-theoretic, and topological. We show how these results can be applied in a straightforward and constructive manner to define homeomorphisms between two given spaces of $p$-adic integers, $\mathbb{Z}_{p}$ and $\mathbb{Z}_{q}$, endowed with their corresponding standard non-archimedean metric topologies.