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Statement

Context (p. 343). The paper recalls Erdős's theorem, from Stöhr's survey (its reference [6]) and Halberstam and Roth (reference [5], p. 89), that every infinite Sidon set $\mathcal A$ satisfies

$$\liminf_{n\to+\infty}A(n)n^{-1/2}(\log n)^{1/2}<+\infty, \qquad(11.1)$$

and notes that this implies $\limsup_{i\to+\infty}(s_{i+1}-s_i)(\log s_i)^{-1}>0$ (11.2) for the sumset $\mathcal S_{\mathcal A}=\{s_1,s_2,\ldots\}$. The authors conjecture that this limsup is $+\infty$, and say this seems very difficult.

Theorem 5 (p. 343, quoted). "There is a positive absolute constant $c_4$ such that if $\mathcal A$ is a finite Sidon set with $|\mathcal A|\ge2$ and we write $\mathcal S_{\mathcal A}=\{s_1,s_2,\ldots,s_u\}$, then we have"

$$\max_{1\le i\le u-1}(s_{i+1}-s_i)>c_4\log|\mathcal A|.$$

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; (11.1), (11.2) and the statement on p. 343, the proof on pp. 343--345. The edition read is identified on the source card.

Read depth. Claims checked: (11.1), (11.2) and the statement 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. 343--345, adapting the proof of (11.1) to finite sets. After a translation, $\min\mathcal A=1$; with $N=[a_v^{1/2}]$ for the largest element $a_v$, the argument of Halberstam and Roth (pp. 89--90) gives some $l\le N$ with $A(lN)\ll(lN/\log N)^{1/2}$. Then at most $A(lN)^2$ sums lie in $[1,lN]$ while the sums span from $2$ to beyond $lN$, so some gap exceeds a constant times $\log N\gg\log|\mathcal A|$.

Dependencies

The argument behind (11.1) in Halberstam and Roth, Sequences, pp. 89--90.

Bears on

• Problem 158: the bound (11.1) recalled here answers the problem yes for Sidon sets, and the paper's Problem 9 asks whether it extends to sets with at most two representations. Theorem 5 itself is about finite Sidon sets and decides no case of the problem.