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Claim. The answer to Problem 785 is yes. The claimed result is the theorem of J.-H. Fang and Y.-G. Chen, On additive complements, in the form the site's commentary records: if $A,B\subseteq\mathbb N$ are infinite, $A+B$ contains every large integer and

$$\limsup_{x\to\infty}\frac{A(x)B(x)}{x}<\frac54,$$

then $A(x)B(x)-x\to\infty$. The problem's hypothesis $A(x)B(x)\sim x$ gives $\limsup=1$, so the theorem contains the problem's statement, which Sárközy and Szemerédi had proved (their claim page); its content is the weaker hypothesis. The same authors raised the threshold to $3-\sqrt3$ (their 2014 claim page) and showed in On additive complements. II (Proc. Amer. Math. Soc. (2011), 881--883, the site's [ChFa11]) that $3/2$ cannot be exceeded: there are such $A,B$ with $\limsup A(x)B(x)/x=3/2$ and $A(x)B(x)-x=1$ for infinitely many $x$. Chen's conjecture that $3/2$ is the true threshold is the subject of the proof claim of van Doorn, Liu and Tang. The library holds no copy; the statement follows the site's commentary and the formal-conjectures variant erdos_785.variants.chen_fang_limsup, which states the $3-\sqrt3$ form without proof, and the publication data follow the Crossref record.

Depends on. Nothing in this wiki.

Acceptance. Refereed: Proc. Amer. Math. Soc. 138 (2010), no. 6, 1923--1927, doi:10.1090/S0002-9939-10-10205-6; the Crossref record dates the publication 5 February 2010, this page's date. Reviewed: the site's curator, Thomas Bloom, labels the problem PROVED (LEAN) and credits this theorem, as [ChFa10], in the problem page's commentary; the curator had no part in the result. No review by this project is recorded.