Upper bounds for finite additive $2$-bases

Abstract:

For a positive integer $N$, a set $\mathcal {A}\subset [0,N]\cap \mathbb {Z}$ is called a $2$-basis for $N$ if every integer $n\in [0,N]$ can be represented as $n=a+b$, where $a, b\in \mathcal {A}$. In this paper, we give a lower bound estimate for the cardinality of an additive $2$-basis for $N$, as $N\to \infty$, which improves the existing results on this topic.