erdos_38

Does there exist $B \subset \mathbb{N}$ which is not an additive basis, but is such that for every set $A \subseteq \mathbb{N}$ of Schnirelmann density $\alpha$ and every $N$ there exists $b \in B$ such that \[ \lvert (A \cup (A+b)) \cap \{1, \ldots, N\} \rvert \geq (\alpha + f(\alpha)) N \] where $f(\alpha) > 0$ for $0 < \alpha < 1$?

theorem erdos_38 : answer(sorry) ↔
    ∃ B : Set ℕ, ¬ B.IsAddBasis ∧ ∃ f : ℝ → ℝ, (∀ α, 0 < α → α < 1 → f α > 0) ∧
      ∀ (A : Set ℕ) (N : ℕ),
        let α

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