Suppose that S is a set of positive integers with the property that S+S -- that is, all sums of the form s_(1)+s_(2) for

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.

For $S\subset\mathbb N$, put $S+S=\{a+b:a,b\in S\}$ and \[r_S(n)=\#\{(a,b)\in S^2:a\le b,\ a+b=n\}.\] Repeated summands are allowed. The set $S$ is an asymptotic additive basis of order two if $r_S(n)\ge1$ for every sufficiently large integer $n$. Counting ordered representations instead changes this count by at most a factor of two, so boundedness is unchanged.

Statement recorded in the supplied source.

Must every asymptotic additive basis $S\subset\mathbb N$ of order two satisfy \[\limsup_{n\to\infty}r_S(n)=+\infty?\] Equivalently, can there be such an $S$ and a finite constant $C$ with $r_S(n)\le C$ for every $n\in\mathbb N$? The conjectured answer to the existence question is negative. [S2]

Short English statement

Must the number of ways to write an integer as the sum of two elements of an asymptotic additive basis become arbitrarily large?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3100055 (AMR-030-0055). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Melvyn B. Nathanson, Generalized additive bases, König's lemma, and the Erdős--Turán conjecture (2003), Section1, unordered representation function, asymptotic bases, and displayed Erdős--Turán conjecture, pages1--2. https://arxiv.org/abs/math/0302155.