Answers to two questions posed by Farhi concerning additive bases

TL;DR

This paper addresses two questions posed by Farhi about improving bounds on the order of additive bases after removing finite subsets, providing definitive answers and new bounds.

Contribution

The paper provides answers to Farhi's questions, offering improved bounds on the order of additive bases after finite subset removal.

Findings

Answered both questions posed by Farhi.

Established new upper bounds for the order of A extbackslash X.

Confirmed the optimality of certain bounds.

Abstract

Let A be an asymptotic basis for N and X a finite subset of A such that A\X is still an asymptotic basis. Farhi recently proved a new batch of upper bounds for the order of A\X in terms of the order of A and a variety of parameters related to the set X. He posed two questions concerning possible improvements to his bounds. In this note, we answer both questions.