Essentialité dans les bases additives Original Research Article

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−P does not 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 radicalʹs 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 complete the result of Deschamps and Grekos about the majoration of essential elements of a basis by showing that for all basis A of order h, the number s of essential elements of A satisfy image where image, and we show that this inequality is best possible.