erdos_899

Let $A\subseteq\mathbb{N}$ be an infinite set such that $|A\cap \{1, ..., N\}| = o(N)$. Is it true that $$ \limsup_{N\to\infty}\frac{|(A - A)\cap \{1, ..., N\}|}{|A \cap \{1, ..., N\}|} = \infty? $$ The answer is yes, proved by Ruzsa [Ru78]. [Ru78] Ruzsa, I. Z., _On the cardinality of {$A+A$}\ and {$A-A$}_. (1978), 933--938.

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

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