Erdős problem 899 · problems.science

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

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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.

FormalConjectures/ErdosProblems/899.leanErdos899.erdos_8995 linesExact file

1True ↔2  ∀ (A : Set ℕ),3    A.Infinite →4      Filter.Tendsto (fun N => ↑(A ∩ Set.Icc 1 N).ncard / ↑N) Filter.atTop (nhds 0) →5        Filter.limsup (fun N => ↑((A - A) ∩ Set.Icc 1 N).ncard / ↑(A ∩ Set.Icc 1 N).ncard) Filter.atTop = ⊤

SolvedStatement only, no proof

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