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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. III, in the form the site's commentary records: if $A,B\subseteq\mathbb N$ are infinite, $A+B$ contains every large integer and

$$\limsup_{x\to\infty}\frac{A(x)B(x)}{x}<3-\sqrt3\approx1.268,$$

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, raised from the $5/4$ of the authors' first paper (their 2010 claim page). The threshold cannot pass $3/2$, by the authors' construction of 2011 recorded on that page, and 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 it without proof, and the publication data follow the Crossref record.

Depends on. Nothing in this wiki.

Acceptance. Refereed: J. Number Theory 141 (2014), 83--91, doi:10.1016/j.jnt.2014.01.027; the Crossref record dates the issue to August 2014, filled to the first of the month for this page's name. Reviewed: the site's curator, Thomas Bloom, labels the problem PROVED (LEAN) and credits the improvement of $5/4$ to $3-\sqrt3$, as [ChFa14], in the problem page's commentary; the curator had no part in the result. No review by this project is recorded.