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Statement

Setting (p. 337). For $n\in\mathbb N$, $H(n)$ is the smallest positive integer $H$ for which some Sidon set $\mathcal A\subset\{1,2,\ldots,n\}$ satisfies $\{i+1,i+2,\ldots,i+H\}\cap\mathcal S_{\mathcal A}\ne\varnothing$ for $i=0,1,\ldots,n$, where $\mathcal S_{\mathcal A}=\mathcal A+\mathcal A$.

Theorem 3 (p. 337, quoted). "For $n\in\mathbb N$, $n>n_0$ we have $H(n)\le 3n^{1/2}$."

The authors remark (p. 337) that almost certainly $H(n)=o(n^{1/2})$, which they could not prove, and that perhaps even $H(n)=o(n^{\varepsilon})$ for all $\varepsilon>0$. Both are conjectures.

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 definition and statement on p. 337, the proof on pp. 337--338. The edition read is identified on the source card.

Read depth. Claims checked: the definition, the statement and the remarks were read clause by clause on the page images of the journal print. The proof was read but not checked step by step.

Proof pointer

Pp. 337--338, an explicit construction modelled on Erdős's (the paper's references [6] and [5, p. 90]), with some details left to the reader. Take $p$ the least prime with $2(p-2)p>n$, so $p=(1+o(1))(n/2)^{1/2}$, put $a_k=2(k-1)p+r(k^2,p)$ for $k=1,\ldots,p-1$, with $r(k^2,p)$ the least nonnegative residue of $k^2$ modulo $p$, and keep the $a_k\le n$. The set is Sidon, contains $a_1=1$, and the sums $a_k+a_1$ step by less than $3p$, which for large $n$ is below $3n^{1/2}$.

Dependencies

None beyond the cited construction.

Bears on

No Erdős problem page of the corpus consumes this theorem. The set it builds is not shown to be a maximal Sidon set, so it says nothing on Problem 156.