Erdős Problem 38

Reference: erdosproblems.com/38

theorem Erdos38.erdos_38 : sorry ↔ ∃ (B : Set ℕ), ¬B.IsAddBasis ∧ ∃ (f : ℝ → ℝ), (∀ (α : ℝ), 0 < α → α < 1 → f α > 0) ∧ ∀ (A : Set ℕ) (N : ℕ), have α := schnirelmannDensity A; ∃ b ∈ B, ↑(Set.Ioc 0 N ∩ (A ∪ (A + {b}))).ncard ≥ (α + f α) * ↑N

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 $0 < \alpha < 1$?