Sur le nombre d'éléments exceptionnels d'une base additive

Abstract

In a paper recently published in Crelle's Journal, Deschamps and Grekos [B. Deschamps et G. Grekos, Estimation du nombre d'exceptions à ce qu'un ensemble de base privé d'un point reste un ensemble de base, J. reine angew. Math. 539 (2001), 45–53.] study asymptotically (when h tends to infinity) the quantity E(h), introduced by Erdős and Graham, and defined as the maximal number of elements which are necessary to the basicity of an additive basis of order h. They show that the maximal order of this function is (h/log h)½. The aim of this article is to show that the E function does not have oscillations but, to the contrary, does possess a regular asymptotic behaviour, that we determine explicitly. More precisely, we prove that E(h) ~ 2(h/log h)½.