A simple proof that any additive basis has only finitely many essential subsets

TL;DR

This paper provides a straightforward proof that any additive basis has only finitely many essential subsets, clarifying a previously established but complex theorem in additive number theory.

Contribution

It offers a simple, accessible proof of a known theorem that finite additive bases possess only finitely many essential subsets.

Findings

Proves finiteness of essential subsets for any additive basis

Simplifies the proof of a key theorem in additive number theory

Enhances understanding of the structure of additive bases

Abstract

Let $A$ be an additive basis. We call ``essential subset'' of $A$ any finite subset $P$ of $A$ such that $A \setminus P$ is not an additive basis and that $P$ is minimal (for the inclusion order) to have this property. A recent theorem due to B. Deschamps and the author states that any additive basis has only finitely many essential subsets (see ``Essentialit\'e dans les bases additives, J. Number Theory, 123 (2007), p. 170-192''). The aim of this note is to give a simple proof of this theorem.