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

Abstract

Let $A$ be an additive basis of order $h$ and $X$ be a finite nonempty subset of $A$ such that the set $A \setminus X$ is still a basis. In this article, we give several upper bounds for the order of $A \setminus X$ 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 $d := \frac{\diam(X)}{\gcd\{x - y | x, y \in X\}}$, we show that the order of $A \setminus X$ is bounded above by $(\frac{h (h + 3)}{2} + d \frac{h (h - 1) (h + 4)}{6})$. As a consequence, we deduce that if $X$ is an arithmetic progression of length $\geq 3$, 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.