A metric for the set of all additive basis

TL;DR

This paper introduces a new topological metric for the set of all additive bases, enabling the analysis of their properties and stability under small modifications in additive number theory.

Contribution

It proposes a novel metric for additive bases that facilitates studying their properties and stability, a new approach in additive number theory.

Findings

The metric can identify bases that remain unchanged when adding certain integers.

It allows analysis of the stability of additive bases under small perturbations.

Provides a new tool for topological study of additive bases.

Abstract

The aim of this article is to present a topological tool for the study of additive basis in additive number theory. It will be proposal a metric for the set of all additive basis, in which it will be possible to study properties of some additive bases studying basis near the chosen basis. This metric allow, for example, detect some additive basis in which we can add some integers in it without change its order.