Upper bounds for the order of an additive basis obtained by removing a finite subset of a given basis

Let A be an additive basis of order h and X be a finite nonempty subset of A such that the set A∖X is still a basis. In this article, we give several upper bounds for the order of A∖X in function of the order h of A and some parameters related to X and A. If the parameter in question is the cardinality of X, Nathanson and Nash already obtained some of such upper bounds, which can be seen as polynomials in h with degree (|X|+1). Here, by taking instead of the cardinality of X the parameter defined by , we show that the order of A∖X is bounded above by . As a consequence, we deduce that if X is an arithmetic progression of length ⩾3, then the upper bounds of Nathanson and Nash are considerably improved. Further, by considering more complex parameters related to both X and A, we get upper bounds which are polynomials in h with degree only 2.