---

Claim. The answer to Problem 785 is yes: if $A,B\subseteq\mathbb N$ are infinite, $A+B$ contains every large integer and $A(x)B(x)\sim x$, then $A(x)B(x)-x\to\infty$. The claimed result is the theorem of A. Sárközy and E. Szemerédi, On a problem in additive number theory, which proves more: under these hypotheses the excess $A(x)B(x)-x$ cannot be $o(A(x))$. The library holds no copy; the theorem follows the site's commentary and Ruzsa's introduction (library home of Ruzsa's paper, ruzsa_2017_exact_additive_complements, whose introduction also records that the result was announced in the 1966 edition of Halberstam and Roth's Sequences). Erdős and Danzer conjectured the statement after Danzer showed that such exact additive complements exist, against Hanani's conjecture that they do not. Sárközy and Szemerédi also conjectured that the excess can be $O(\min\{A(x),B(x)\})$; Chen and Fang disproved that conjecture (library card; claim page). Their theorems under the weaker hypotheses $\limsup A(x)B(x)/x<5/4$ and $<3-\sqrt3$ (2010, 2014) and the sharper lower bound of Ruzsa are later results with their own pages, not part of this claim.

Depends on. Nothing in this wiki.

Acceptance. Refereed: Acta Math. Hungar. 64 (1994), no. 3, 237--245, doi:10.1007/BF01874252; the Crossref record dates the issue to September 1994, filled to the first of the month for this page's name. Reviewed: the site's curator, Thomas Bloom, credits the affirmative answer to Sárközy and Szemerédi in the problem page's commentary and labels the problem PROVED (LEAN) on the page last edited 7 March 2026, and the formal-conjectures catalog's statement file for the problem names them as the source of the proof; the community database lists the problem as proved as of its last update on 2026-03-06. Nothing here rests on a review by this project.