erdos_741.density

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

theorem erdos_741.density : answer(sorry) ↔ ∀ A : Set ℕ, HasPosDensity (A + A) → ∃ A₁ A₂,
    A = A₁ ∪ A₂ ∧ Disjoint A₁ A₂ ∧ HasPosDensity (A₁ + A₁)
    ∧ HasPosDensity (A₂ + A₂)

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