The Erdos-Turan conjecture on additive bases

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Here use $\mathbb N_0=\{0,1,2,\ldots\}$. For $B\subseteq\mathbb N_0$ define the ordered representation count \[ r_B(k)=\#\{(a,b)\in B^2:a+b=k\}\qquad(k\in\mathbb N_0). \] An additive basis of order two, in the source's everywhere-positive convention, satisfies $r_B(k)>0$ for every $k\in\mathbb N_0$. Repeated summands are allowed, and $(a,b),(b,a)$ are counted separately when $a\ne b$.

Statement recorded in the supplied source.

Does every such additive basis have an unbounded representation function? Equivalently, must \[ \forall M\in\mathbb N\quad\exists k\in\mathbb N_0:\quad r_B(k)>M \] hold whenever $r_B(k)>0$ for all $k$? [S2]

Short English statement

Must the number of ordered representations grow without bound for any set that represents every nonnegative integer as a sum of two of its elements?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3363 (OPG-367). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, The Erdos-Turan conjecture on additive bases, OPG-367. Original problem statement, full mathematical discussion, bibliography and comments. https://www.openproblemgarden.org/op/the_erdos_turan_conjecture_on_additive_bases.