A NOTE ON THE ERDŐS–GRAHAM THEOREM

Abstract

Let ${\mathcal{A}}=\{a_{1}<a_{2}<\cdots \,\}$ be a set of nonnegative integers. Put $D({\mathcal{A}})=\gcd \{a_{k+1}-a_{k}:k=1,2,\ldots \}$. The set ${\mathcal{A}}$ is an asymptotic basis if there exists $h$ such that every sufficiently large integer is a sum of at most $h$ (not necessarily distinct) elements of ${\mathcal{A}}$. We prove that if the difference of consecutive integers of ${\mathcal{A}}$ is bounded, then ${\mathcal{A}}$ is an asymptotic basis if and only if there exists an integer $a\in {\mathcal{A}}$ such that $(a,D({\mathcal{A}}))=1$.