Erdős Problem 38

Reference:

• erdosproblems.com/38
• [Er56](Erdős, P., Problems and results in additive number theory. Colloque sur la Théorie des Nombres, Bruxelles, 1955 (1956), 127-137.)

theorem Erdos38.erdos_38 : True ↔ ∃ (B : Set ℕ), ¬B.IsWeakAddBasis ∧ ∃ (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 $0 < \alpha < 1$?

Note: here Erdős seems to use a slightly weaker notion of an additive basis (see [Er56] at the top of page 135). In particular, for this problem, a set is an additive basis of order $k$ if every natural number can be written as a sum of at most $k$ elements of the set, rather than as a sum of precisely $k$ elements.