Erdős Problem 741

References:

• erdosproblems.com/741
• [Er94b] Erdős, Paul, Some problems in number theory, combinatorics and combinatorial geometry. Math. Pannon. (1994), 261-269.

theorem Erdos741.erdos_741.density : sorry ↔ ∀ (A : Set ℕ), (A + A).HasPosDensity → ∃ (A₁ : Set ℕ) (A₂ : Set ℕ), A = A₁ ∪ A₂ ∧ Disjoint A₁ A₂ ∧ (A₁ + A₁).HasPosDensity ∧ (A₂ + A₂).HasPosDensity

Let A ⊆ ℕ be a set such that A + A has positive density. Can one always decompose A as a disjoint union of two subsets A₁ and A₂ such that both A₁ + A₁ and A₂ + A₂ have positive density?

theorem Erdos741.erdos_741.basis : sorry ↔ ∀ (A : Set ℕ), (A ∪ {0}).IsAddBasisOfOrder 2 → ∃ (A₁ : Set ℕ) (A₂ : Set ℕ), A = A₁ ∪ A₂ ∧ Disjoint A₁ A₂ ∧ ¬(IsSyndetic A₁ ∧ IsSyndetic A₂)

Let A ⊆ ℕ be a basis of order 2. Can one always decompose A as a disjoint union of two subsets A₁ and A₂ such that A₁ + A₁ and A₂ + A₂ cannot both have bounded gaps?