erdos_331.variant.ruzsa

Ruzsa suggests that a non-trivial variant of this problem arises if one imposes the stronger condition that $|A \cap \{1,\dots,N\}| \sim c_A N^{1/2}$ for some constant $c_A>0$, and similarly for $B$.

theorem erdos_331.variant.ruzsa :
    answer(sorry) ↔
      ∀ A B : Set ℕ,
      (∃ c_A > 0, (fun (n : ℕ) ↦ (count A n : ℝ)) ~[atTop] (fun (n : ℕ) ↦ c_A * (n : ℝ) ^ (1 / 2 : ℝ))) →
      (∃ c_B > 0, (fun (n : ℕ) ↦ (count B n : ℝ)) ~[atTop] (fun (n : ℕ) ↦ c_B * (n : ℝ) ^ (1 / 2 : ℝ))) →
      { s : ℕ × ℕ × ℕ × ℕ | let ⟨a₁, a₂, b₁, b₂⟩

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