Source labels openChecked July 26, 2026

Erdős Problem 741: Lower

Let $A\subseteq \mathbb{N}$ be such that $A+A$ has positive lower 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 lower density?

Combinatorics AMS 5

Mathematical statement

Let $A\subseteq \mathbb{N}$ be such that $A+A$ has positive lower 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 lower density?

Statement source: Erdős Problems statement material

Statement terms: Source-specific

Source-specific terms. Therefore does not assert reuse rights beyond attributed display.

Statement artifacts, not proofs

These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.

Pinned Lean formulation 1

erdos_741.variants.lower

Canonical source

Complete statement target, proof intentionally absentLean 4

theorem erdos_741.variants.lower : answer(sorry) ↔ ∀ A : Set ℕ, 0 < lowerDensity (A + A) → ∃ A₁ A₂,    A = A₁ ∪ A₂ ∧ Disjoint A₁ A₂ ∧ 0 < lowerDensity (A₁ + A₁)    ∧ 0 < lowerDensity (A₂ + A₂) := by  sorry

Statement source: Formal Conjectures
Lean version: v4.27.0
Placeholder: Present; no proof artifact
Source evidence: Pinned source index
Fidelity review: Community formulation

References

Pinned Formal Conjectures sourceformalization erdosproblems.com/741background