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Statement

Setting (pp. 329--330). A set $\mathcal A\subset\mathbb N$ is a Sidon set when the sums $a+a'$ with $a,a'\in\mathcal A$, $a\le a'$, are distinct. The paper writes $\mathcal S_{\mathcal A}=\mathcal A+\mathcal A$ and, for $\mathcal X\subset\mathbb N$ and $d\in\mathbb N$, $\mathcal B(\mathcal X,d)=\{x\in\mathcal X: x-d\notin\mathcal X\}$, the elements of $\mathcal X$ with no predecessor at distance $d$ in $\mathcal X$. For every finite $\mathcal A$ and every $d$ the trivial bound (3.1) (p. 330) is $|\mathcal B(\mathcal S_{\mathcal A},d)|\le|\mathcal S_{\mathcal A}|\le|\mathcal A|^2$.

Theorem 1 (p. 330, quoted). "There is a positive constant $c_1$ such that for every finite Sidon set $\mathcal A$ and all $d\in\mathbb N$ we have $|\mathcal B(\mathcal S_{\mathcal A},d)|>c_1|\mathcal A|^2$."

The paper draws the case $d=1$ (p. 330): writing $\mathcal S_{\mathcal A}$ as the union of $t=|\mathcal B(\mathcal S_{\mathcal A},1)|$ maximal blocks of consecutive integers, the number $t$ of blocks satisfies $t\gg|\mathcal A|^2$. With (3.1), the theorem is sharp up to the constant.

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; the definitions on pp. 329--330, the statement on p. 330. The edition read is identified on the source card.

Read depth. Claims checked: the definitions and the statement were read clause by clause on the page images of the journal print.

Proof pointer

None in the paper. The authors say (p. 331) that Theorems 1 and 2 can be proved similarly, that the proof of Theorem 1 is the simpler, and they give only the proof of Theorem 2.

Dependencies

None stated.

Bears on

• Problem 864: the theorem needs a Sidon set, and a set of Problem 864 with a repeated sum is not one. The paper's Problem 5 asks whether a bound of this kind persists for nearly Sidon sets, a class that by that page's note contains the sets of Problem 864 of growing size. The theorem gives no bound on the size of such a set.