Erdős problem 245 · problems.science

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

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FormalConjectures/ErdosProblems/

245.lean

Retained formal statement2 of 2

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

FormalConjectures/ErdosProblems/245.leanErdos245.erdos_245.variants.two4 linesExact file

1∀ (A : Set ℕ),2  A.Infinite →3    Filter.Tendsto (fun N => ↑(A ∩ Set.Icc 1 ⌊N⌋₊).ncard / N) Filter.atTop (nhds 0) →4      2 ≤ Filter.limsup (fun N => ↑((A + A) ∩ Set.Icc 1 ⌊N⌋₊).ncard / ↑(A ∩ Set.Icc 1 ⌊N⌋₊).ncard) Filter.atTop

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