erdos_868.parts.ii

Let $A$ be an additive basis of order $2$, let $f(n)$ denote the number of ways in which $n$ can be written as the sum of two elements from $A$. If $f(n) > \epsilon \log n$ for large $n$ and an arbitrary fixed $\epsilon > 0$, then must $A$ contain a minimal additive basis of order $2$?

theorem erdos_868.parts.ii :
    answer(sorry) ↔ ∀ᵉ (A : Set ℕ) (ε > 0), A.IsAsymptoticAddBasisOfOrder 2 →
      (∀ᶠ (n : ℕ) in atTop, ε * Real.log n < ncard_add_repr A 2 n) → ∃ B ⊆ A,
      B.IsAsymptoticAddBasisOfOrder 2 ∧ ∀ b ∈ B, ¬(B \ {b}).IsAsymptoticAddBasisOfOrder 2

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