Essentialit\'e dans les bases additives

Abstract

In this article we study the notion of essential subset of an additive basis, that is to say the minimal finite subsets $P$ of a basis $A$ such that $A \setminus P$ doesn't remains a basis. The existence of an essential subset for a basis is equivalent for this basis to be included, for almost all elements, in an arithmetic non-trivial progression. We show that for every basis $A$ there exists an arithmetic progression with a biggest common difference containing $A$. Having this common difference $a$ we are able to give an upper bound to the number of essential subsets of $A$: this is the radical's length of $a$ (in particular there is always many finite essential subsets in a basis). In the case of essential subsets of cardinality 1 (essential elements) we introduce a way to "dessentialize" a basis. As an application, we definitively improve the earlier result of Deschamps and Grekos giving an upper bound of the number of the essential elements of a basis. More precisely, we show that for all basis $A$ of order $h$, the number $s$ of essential elements of $A$ satisfy $s\leq c\sqrt{\frac{h}{\log h}}$ where $c=30\sqrt{\frac{\log 1564}{1564}}\simeq 2,05728$, and we show that this inequality is best possible.