Coset topologies on $\mathbb{Z}$ and arithmetic applications

TL;DR

This paper introduces a unified construction of topologies on integers, including Golomb and Kirch topologies, connecting them to profinite completions and exploring their applications in number theory.

Contribution

It presents a general framework for many integer topologies and links them to the profinite completion, offering new insights and applications in number theory.

Findings

Golomb and Kirch topologies are part of a connected, Hausdorff family on FZ.

The construction relates integer topologies to closed sets in FZ.

Applications to number theory are discussed.

Abstract

We provide a construction which covers as special cases many of the topologies on integers one can find in the literature. Moreover, our analysis of the Golomb and Kirch topologies inserts them in a family of connected, Hausdorff topologies on $\mathbb{Z}$, obtained from closed sets of the profinite completion $\hat{\mathbb{Z}}$. We also discuss various applications to number theory.