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Statement

Setting. $\mathcal S_{\mathcal A}=\mathcal A+\mathcal A$ and $\mathcal B(\mathcal X,d)=\{x\in\mathcal X:x-d\notin\mathcal X\}$, as in Theorem 1. By (2.1) (p. 329), $|\mathcal S_{\mathcal A}|\le\binom{|\mathcal A|}2+|\mathcal A|$ for every finite $\mathcal A$, with equality exactly for Sidon sets (p. 330).

Problem 5 (p. 346). The authors propose extending the problems of the paper to "nearly" Sidon sets (their quotation marks). In particular they ask whether every finite set $\mathcal A$ with

$$|\mathcal S_{\mathcal A}|=\left(\tfrac12+o(1)\right)|\mathcal A|^2$$

must have $|\mathcal B(\mathcal S_{\mathcal A},d)|$ large, perhaps $\gg|\mathcal A|^2$, for all $d\in\mathbb N$. They add that the method of the proof of Theorem 2 cannot be adapted to this problem. The paper gives no result on it.

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, p. 346. The edition read is identified on the source card.

Read depth. Claims checked: the problem was read clause by clause on the page images of the journal print. A question has no proof to check; the note below is the corpus's own.

Note

Count representations $s=a+a'$ with $a\le a'$, as the paper does. If a set of $m$ elements has exactly one sum $s_0$ with $r\ge2$ representations, and every other sum has one, then $|\mathcal S_{\mathcal A}|=\binom{m+1}2-(r-1)$. The pairs representing $s_0$ are disjoint, so $r\le\lfloor(m+1)/2\rfloor$, and the set meets the hypothesis of Problem 5 as $m\to\infty$.

Dependencies

None.

Bears on

• Problem 864: by the note above, sets of Problem 864 whose size tends to infinity meet the nearly Sidon hypothesis of Problem 5. Problem 5 asks about the block structure of the sumset, not about the size of the set, and the paper resolves neither question.