The board / additive combinatorics / #899
Erdős Problem #899
PROVED LEANdatabase updated 2026-08-23
The statement as posed is true, and a proof exists. A solution has also been checked by machine, in Lean.
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Let \(A\subseteq\mathbb{N}\) be an infinite set such that \(|A\cap \{1, ..., N\}| = o(N)\). Is it true that \[ \limsup_{N\to\infty}\frac{|(A - A)\cap \{1, ..., N\}|}{|A \cap \{1, ..., N\}|} = \infty? \] The answer is yes, proved by Ruzsa [Ru78]. [Ru78] Ruzsa, I. Z., _On the cardinality of {\(A+A\)}\ and {\(A-A\)}_. (1978), 933--938.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved. The authoritative wording is on erdosproblems.com/899.Prizenone
Topicsadditive combinatorics
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/899
Attempts
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