erdos_489

Let $A\subseteq \mathbb{N}$ be a set such that $\lvert A\cap [1,x]\rvert=o(x^{1/2})$. Let $B=\{ n\geq 1 : a\nmid n\textrm{ for all }a\in A\}$. If $B=\{b_1 < b_2 < \cdots\}$ then is it true that $$\lim_{x \to \infty} \frac{1}{x}\sum_{b_i < x}(b_{i+1}-b_i)^2$$ exists (and is finite)? For example, when $A=\{p^2: p\textrm{ prime}\}$ then $B$ is the set of squarefree numbers, and the existence of this limit was proved by Erdős. See also [208].

theorem erdos_489 : answer(sorry) ↔
    ∀ (A : Set ℕ),
      (fun x : ℕ => (((Finset.Icc 1 x).filter (· ∈ A)).card : ℝ)) =o[atTop]
        (fun x : ℕ => (x : ℝ).sqrt) →
      (sievedSet A).Infinite →
      ∃ L : ℝ, Tendsto (fun x : ℕ => GapSumSq A x / (x : ℝ)) atTop (𝓝 L)

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