The board / additive combinatorics / #741
Erdős Problem #741
SOLVED LEANdatabase updated 2025-08-31
Resolved in some way other than a straight proof or disproof — usual for open-ended questions, or for multi-part problems whose parts went different ways. A solution has also been checked by machine, in Lean.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
Let \(A\subseteq \mathbb{N}\) be such that \(A+A\) has positive upper density. Can one always decompose \(A=A_1\sqcup A_2\) such that \(A_1+A_1\) and \(A_2+A_2\) both have positive upper density? This was proved by the DeepMind prover agent.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 3 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/741.Prizenone
Topicsadditive combinatorics
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/741
Attempts
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