1. FormalConjectures
  2. ErdosProblems
  3. «899»
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Erdős Problem 899

Reference: erdosproblems.com/899

open Filter Set open scoped Pointwise Topology namespace Erdos899 open Erdos899

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.

@[category research solved, AMS 5] theorem declaration uses 'sorry'erdos_899 : answer(True) ↔ ∀ (A : Set ℕ), A.Infinite → Tendsto (fun N => (A ∩ Icc 1 N |>.ncard : ℝ) / N) atTop (𝓝 0) → atTop.limsup (fun N => ((A - A : Set ℕ) ∩ Icc 1 N |>.ncard : EReal) / (A ∩ Icc 1 N).ncard) = ⊤ := ⊢ True ↔ ∀ (A : Set ℕ), A.Infinite → Tendsto (fun N => ↑(A ∩ Icc 1 N).ncard / ↑N) atTop (𝓝 0) → limsup (fun N => ↑((A - A) ∩ Icc 1 N).ncard / ↑(A ∩ Icc 1 N).ncard) atTop = ⊤ All goals completed! 🐙 end Erdos899