Let $A\subseteq \mathbb{N}$ be an infinite set such that $|A\cap \{1,\dots ,N\}|=o(N)$. Is it true that$$\underset{N\to \infty }{\mathrm{limsup}}\frac{|(A+A)\cap \{1,\dots ,N\}|}{|A\cap \{1,\dots ,N\}|}\ge 3?$$

AMS 5 11 · solved (literature)

theorem erdos_245 :
    answer(True) ↔ ∀ (A : Set ℕ), A.Infinite →
      atTop.Tendsto (fun N ↦ (A ∩ Icc 1 ⌊N⌋₊ |>.ncard : ℝ) / N) (𝓝 0) →
      3 ≤ atTop.limsup
        fun N : ℝ ↦ ((A + A) ∩ Icc 1 ⌊N⌋₊ |>.ncard : EReal)
          / (A ∩ Icc 1 ⌊N⌋₊).ncard