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Statement

Setting (p. 1). $B_2[g]$ is the class of sets $A\subset\mathbb N$ such that for every $n\in\mathbb N$ the equation $a+a'=n$ with $a,a'\in A$ and $a\le a'$ has at most $g$ solutions, and $A(x)=\#\{a\le x : a\in A\}$.

The introduction first recalls Erdős's theorem, cited through Stöhr's survey (reference [6]), that every infinite Sidon sequence has $\liminf_{x\to\infty}A(x)/x^{1/2}=0$. It then states (p. 1, quoted): "It is conjectured that $\liminf_{x\to\infty}\frac{A(x)}{x^{1/2}}=0$ for any infinite $B_2[g]$ sequence, but it is unknown even for $g=2$."

Scope

This is a conjecture the paper records, naming no source for it; the paper neither proves nor attacks it. Its Theorem 1 concerns the limit superior and gives no information on the limit inferior.

Read depth. Claims checked: the sentence and the definitions before it were read clause by clause on p. 1 of the print.

Source. J. Cilleruelo and C. Trujillo, Infinite $B_2[g]$ sequences, Israel Journal of Mathematics 126 (2001), 263--267, doi:10.1007/BF02784156, read in the author-typeset version named on the source card.

Bears on. #158: the conjecture's case $g=2$ asserts the affirmative answer to the problem's question, with the same convention ($a\le a'$, at most two solutions); the paper records it as open in 2001 and proves nothing about it.