Erdos-Turan conjecture asserts there exist no asymptotic basis $A\subset \naturals_0$ of order $2$ such that its representation function \begin{equation*} r'_{A,2}(n)=\sum_{\substack{a_1+a_2=n\\a_1\leq a_2}} 1 \end{equation*}is bounded.

Alternatively, the question can be phrased as whether there exists a power series $F$ with coefficients $0$ and $1$ such that all coefficients of $F^2$ are greater than $0$ , but are bounded.

If we replace set of nonnegative integers by the set of all integers, then the question was settled by Nathanson[2] in negative, that is, there exists a set $A\subset \integers$ such that $r'_{A,2}(n)=1$ .

References

1

Heini Halberstam and Klaus Friedrich Roth.
Sequences.
Springer-Verlag, second edition, 1983.
Zbl 0498.10001.

2

Melvyn B. Nathanson.
Every function is the representation function of an additive basis for the integers.
arXiv:math.NT/0302091.