On finite additive $2$-bases

Abstract:

For a positive integer $N$, a set $\mathcal {B}$ of integers from $\{0,1,\dots ,N-1\}$ is called an additive $2$-basis for $N$ if every integer $n\in \{0,1,\dots ,N-1\}$ may be represented as the sum of $2$ elements of $\mathcal {B}$. We discuss the methods used to estimate the minimal size of an additive $2$-basis for $N$. We provide new examples to enrich this survey, which give good bounds. For instance, we slightly improve on the current record, from $0.46972$ to $0.46906$.