OPEN This is open, and cannot be resolved with a finite computation.

Let $r\geq 2$ and let $A\subseteq \{1,\ldots,N\}$ be a set of maximal size such that there are at most $r$ solutions to $n=a+b$ with $a\leq b$ for any $n$. (That is, $A$ is a $B_2[r]$ set.)

Similarly, let $B\subseteq \{1,\ldots,N\}$ be a set of maximal size such that there are at most $r$ solutions to $n=a-b$ for any $n$.

If $\lvert A\rvert\sim c_rN^{1/2}$ as $N\to \infty$ and $\lvert B\rvert \sim c_r'N^{1/2}$ as $N\to \infty$ then is it true that $c_r\neq c_r'$ for $r\geq 2$? Is it true that $c_r'<c_r$?

#863: [Er92c]

number theory | sidon sets | additive combinatorics