Let $A\subseteq \mathbb{N}$ be an infinite set such that $|A\cap \{1,\dots ,N\}|=o(N)$. Is it true that$$\underset{N\to \infty }{\mathrm{limsup}}\frac{|(A-A)\cap \{1,\dots ,N\}|}{|A\cap \{1,\dots ,N\}|}=\infty ?$$

AMS 5 · solved (literature)

theorem erdos_899 : answer(True) ↔ ∀ (A : Set ℕ), A.Infinite →
    Tendsto (fun N => (A ∩ Icc 1 N |>.ncard : ℝ) / N) atTop (𝓝 0) →
    atTop.limsup (fun N => ((A - A : Set ℕ) ∩ Icc 1 N |>.ncard : EReal) /
      (A ∩ Icc 1 N).ncard) = ⊤