On Metrics Inducing the F\"urstenberg Topology on the Integers

TL;DR

This paper explores different metrics that induce the F"urstenberg topology on integers, analyzing their properties, computability, and applications to number theory, revealing new insights at the intersection of topology and number theory.

Contribution

It introduces and analyzes various metrics inducing the F"urstenberg topology, connecting them to number theory and establishing new propositions in the field.

Findings

Identified classes of metrics inducing the F"urstenberg topology

Analyzed the computational aspects of these metrics

Established new propositions linking number theory and topology

Abstract

We investigate various classes of metrics on the integers, which induce the F\"urstenberg topology and establish the connection between the metrics and the topology. We analyze the norm-like mappings underlying these metrics, with respect to their efficient computability for natural numbers and the analytic behavior of sequences under those mappings. Subsequently, we give some applications to number theory and establish some new propositions at the intersection of number theory and topology.