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Erdős problem 741

Let A⊆NA\subseteq \mathbb{N} be such that A+AA+A has positive density in the literal sense that its natural density exists and is positive. Can one always decompose A=A1⊔A2A=A_1\sqcup A_2 such that A1+A1A_1+A_1 and A2+A2A_2+A_2 both have positive density in this sense?

No current result

No reviewed Result is current in Vela Mathematics Program. Retained source material is shown below.

Retained declaration

FormalConjectures/ErdosProblems/741.lean

Formal Conjectures

FormalConjectures/ErdosProblems/741.leanErdos741.erdos_741.parts.i4 linesExact file
True ↔  ∀ (A : Set ℕ),    0 < (A + A).upperDensity →      ∃ A₁ A₂, A = A₁ ∪ A₂ ∧ Disjoint A₁ A₂ ∧ 0 < (A₁ + A₁).upperDensity ∧ 0 < (A₂ + A₂).upperDensity
SolvedProof has a holeformal conjecturesexternal proof

The proof uses `sorry`: part of the argument is written but not proved. Lean accepts the file; it does not accept the theorem.

Proof manifests naming this Problem

  • Jayyhk Erdős Leanjayyhk:erdos:741
  • PLBY Lean proofsErdosProblems.Erdos741
  • PLBY Lean proofsErdosProblems.Erdos741b

Reported activity

Work these sources record against this Problem. Source-reported attribution, not reviewed here.

  • AI standalone

    Erdős AI contributions wiki · 31 Mar, 2026

    Machine
    DeepMind prover agent
    Open the source record

  • AI standalone

    Erdős AI contributions wiki · 16 Apr, 2026

    Machine
    DeepMind prover agent
    Open the source record

  • AI standalone

    Erdős AI contributions wiki · 31 Mar, 2026

    Machine
    OpenAI internal model
    Open the source record

  • argument

    VibeMathed

    Machine
    DeepMind prover agent
    People
    Boris Alexeev, Moe Putterman, Mehtaab Sawhney, Mark Sellke, Gregory Valiant
    Reported outcome
    resolved
    Open the source record

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