erdos_245.variants.two

Let $A\subseteq\mathbb{N}$ be an infinite set such that $|A\cap \{1, ..., N\}| = o(N)$. Then $$ \limsup_{N\to\infty}\frac{|(A + A)\cap \{1, ..., N\}|}{|A \cap \{1, ..., N\}|} \geq 2. $$

theorem erdos_245.variants.two (A : Set ℕ) (h_inf : A.Infinite)
    (hf : atTop.Tendsto (fun N ↦ (A.interIcc 1 ⌊N⌋₊ |>.ncard : ℝ) / N) (𝓝 0)) :
    2 ≤ atTop.limsup
      fun N : ℝ ↦ ((A + A).interIcc 1 ⌊N⌋₊ |>.ncard : EReal) / (A.interIcc 1 ⌊N⌋₊).ncard

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