---

Statement

Definition (p. 346). A set $\mathcal A$ is a $B_2[g]$ set if for every $n\in\mathbb N$ the equation $a+a'=n$ with $a\le a'$, $a,a'\in\mathcal A$, has at most $g$ solutions; a Sidon set is a $B_2[1]$, or $B_2$, set.

Problem 9 (pp. 346--347). The authors propose extending the problems and results of the paper to $B_2[g]$ sets, and say this seems very difficult. They illustrate the difficulty (p. 347): the largest Sidon set $\mathcal A\subset\{1,2,\ldots,n\}$ is known to satisfy $|\max|\mathcal A|-n^{1/2}|\ll n^{5/16}$, while no asymptotic formula is known for the largest $B_2[g]$ set in $\{1,2,\ldots,n\}$. Moreover, it is not known whether Erdős's bound (11.1) (p. 343), $\liminf_{n\to+\infty}A(n)n^{-1/2}(\log n)^{1/2}<+\infty$ for every infinite Sidon set, extends to $B_2[2]$ or $B_2[g]$ sets. They put the question as: must every infinite $B_2[2]$ set $\mathcal A$ satisfy

$$\liminf_{n\to+\infty}A(n)n^{-1/2}=0?$$

The paper introduces this displayed question with "In other words"; it is weaker than (11.1) for $B_2[2]$ sets, which would imply it. The paper gives no result on either.

Source. P. Erdős, A. Sárközy, V. T. Sós, On Sum Sets of Sidon Sets, I, J. Number Theory 47 (1994), 329--347, doi:10.1006/jnth.1994.1040; §12, pp. 346--347, with (11.1) on p. 343. The edition read is identified on the source card.

Read depth. Claims checked: the definition, the illustration and the question were read clause by clause on the page images of the journal print. A question has no proof to check.

Dependencies

(11.1), recalled on p. 343; see Theorem 5.

Bears on

• Problem 158: the displayed question is this problem, with $A(n)=|\mathcal A\cap\{1,\ldots,n\}|$ and at most two solutions of $a+a'=n$, $a\le a'$. For Sidon sets, (11.1) answers it yes. The paper poses the $B_2[2]$ case and does not resolve it.