Is the set of odd integers not of the form $2^k+p$ the union of an infinite arithmetic progression and a set of density $0$?

AMS 11 · solved (literature)

theorem erdos_16 :
    answer(False) ↔
      ∃ A B : Set ℕ, Erdos16Set = A ∪ B ∧
        (∃ a d : ℕ, d > 0 ∧ A = { x | ∃ m : ℕ, x = a + m * d }) ∧
        density_zero B