---

Source. Part 3 (Open problems), Problem 2, p. 8, of P. Erdős and M. B. Nathanson, "Partitions of bases into disjoint unions of bases," J. Number Theory 29 (1988), no. 1, 1--9. The edition read is identified on the source card.

Statement

Problem 2 (p. 8). Let $A$ be an asymptotic basis of order $2$, and let $f(n)$ be the number of representations $n=a_i+a_j$ with $a_i,a_j\in A$ and $a_i\le a_j$. By Theorem 3, if $f(n)\ge c\log n$ for some $c>\log^{-1}(4/3)$ and all $n\ge n_0$, then $A$ is the union of two disjoint asymptotic bases of order 2. The paper asks: "Can the condition that $f(n)\geqslant c\log n$ be weakened?" In particular, if only $\lim_{n\to\infty}f(n)=\infty$ is assumed, is $A=A_1\cup A_2$ with $A_1\cap A_2=\emptyset$ and $A_1$, $A_2$ both asymptotic bases of order 2?

The paper gives no result on the question.

Read depth. Claims checked: the problem was read clause by clause on the print.

Dependencies

Theorem 3 (p. 5).

Bears on

• #871: the problem page lists this paper among its references; the question in the second sentence of Problem 2 is the one #871 asks, which the site states with the count $1_A\ast1_A(n)$ of ordered representations in place of the paper's $f(n)$. The paper poses the question and does not answer it.