Does there exist a minimal basis with positive density, say $A\subset \mathbb{N}$, such that for any $n\in A$ the (upper) density of integers which cannot be represented without using $n$ is positive?

AMS 5 11 · open (literature)

theorem erdos_330_statement :
    answer(sorry) ↔ ∃ (A : Set ℕ), ∃ h, MinAsymptoticAddBasisOfOrder A h ∧ A.HasPosDensity ∧
    ∀ n ∈ A, Set.HasPosDensity (UnrepWithout A n h)