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

TL;DR

This paper establishes new upper bounds on the order of an additive basis after removing a finite subset, improving previous bounds by introducing parameters related to the subset's structure.

Contribution

It introduces novel bounds depending on parameters like diameter and gcd, refining earlier polynomial bounds based solely on subset size.

Findings

Upper bounds depend on the diameter and gcd of the subset.

Bounds are improved for subsets that are arithmetic progressions.

Polynomial bounds of degree 2 are achieved for complex parameters.

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.