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Erdős problem 245

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Let A⊆NA\subseteq\mathbb{N} be an infinite set such that ∣A∩{1,...,N}∣=o(N)|A\cap \{1, ..., N\}| = o(N). Is it true that lim sup⁡N→∞∣(A+A)∩{1,...,N}∣∣A∩{1,...,N}∣≥3? \limsup_{N\to\infty}\frac{|(A + A)\cap \{1, ..., N\}|}{|A \cap \{1, ..., N\}|} \geq 3?

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FormalConjectures/ErdosProblems/

245.lean

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Let A⊆NA\subseteq\mathbb{N} be an infinite set such that ∣A∩{1,...,N}∣=o(N)|A\cap \{1, ..., N\}| = o(N). Then lim sup⁡N→∞∣(A+A)∩{1,...,N}∣∣A∩{1,...,N}∣≥2. \limsup_{N\to\infty}\frac{|(A + A)\cap \{1, ..., N\}|}{|A \cap \{1, ..., N\}|} \geq 2.

FormalConjectures/ErdosProblems/245.leanErdos245.erdos_245.variants.two4 linesExact file
∀ (A : Set ℕ),  A.Infinite →    Filter.Tendsto (fun N => ↑(A ∩ Set.Icc 1 ⌊N⌋₊).ncard / N) Filter.atTop (nhds 0) →      2 ≤ Filter.limsup (fun N => ↑((A + A) ∩ Set.Icc 1 ⌊N⌋₊).ncard / ↑(A ∩ Set.Icc 1 ⌊N⌋₊).ncard) Filter.atTop
SolvedStatement only, no proof