representation function

The Erdos-Turan conjecture on additive bases ★★★★

Author(s): Erdos; Turan

Let $B \subseteq {\mathbb N}$. The \emph{representation function} $r_B : {\mathbb N} \rightarrow {\mathbb N}$ for $B$ is given by the rule $r_B(k) = \#\{ (i,j) \in B \times B : i + j = k \}$. We call $B$ an \emph{additive basis} if $r_B$ is never $0$.

\begin{conjecture} If $B$ is an additive basis, then $r_B$ is unbounded. \end{conjecture}

Keywords: additive basis; representation function