Mathematics – Number Theory
Scientific paper
2008-02-08
J. Number Theory, 123 (2007), p. 170-192
Mathematics
Number Theory
24 pages (in french)
Scientific paper
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.
Deschamps Bruno
Farhi Bakir
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