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Claim. Every infinite Sidon set $A\subseteq\mathbb N$, a set in which each $n$ has at most one solution of $a+b=n$ with $a\le b$, satisfies

$$\liminf_{N\to\infty}\frac{\lvert A\cap\{1,\ldots,N\}\rvert(\log N)^{1/2}}{N^{1/2}}<\infty, \qquad\text{hence}\qquad \liminf_{N\to\infty}\frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N^{1/2}}=0.$$

Every Sidon set meets the hypothesis of Problem 158, so the answer is yes for these sets. The theorem is Erdős's, published in A. Stöhr's survey of 1955. Erdős, Sárközy and Sós cite it as (11.1) of their paper, whose Problem 9 poses Problem 158 as its extension to two representations, and the site's remark credits Erdős with the Sidon case. formal-conjectures states it as erdos_158.variants.isSidon, derived from a variant whose proof is left as sorry, so the file carries no formal proof of it.

Covers. The infinite Sidon sets. Sets in which some integer has two representations are not covered.

Depends on. No page of this wiki; the result is the paper's.

Acceptance. Refereed: A. Stöhr, Gelöste und ungelöste Fragen über Basen der natürlichen Zahlenreihe. II, J. Reine Angew. Math. 194 (1955), 111–140, which records Erdős's theorem; the volume carries only its year, so the page name uses 1 January 1955. The site labels the problem OPEN, so its remark is not listed as reviewed.