Erdős Problem 331

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import FormalConjecturesUtil

/-!
# Erdős Problem 331

*Reference:* [erdosproblems.com/331](https://www.erdosproblems.com/331)
-/

open Nat Filter
open scoped Asymptotics

namespace Erdos331

open scoped Classical in
/--
Let $A,B\subseteq \mathbb{N}$ such that for all large $N$$$\lvert A\cap \{1,\ldots,N\}\rvert \gg
N^{1/2}$$and$$\lvert B\cap \{1,\ldots,N\}\rvert \gg N^{1/2}.$$
Is it true that there are infinitely many solutions to $a_1-a_2=b_1-b_2\neq 0$ with $a_1,a_2\in A$
and $b_1,b_2\in B$?

Ruzsa has observed that there is a simple counterexample: take $A$ to be the set of numbers whose
binary representation has only non-zero digits in even places, and $B$ similarly but with non-zero
digits only in odd places. It is easy to see $A$ and $B$ both grow like $\gg N^{1/2}$ and yet for
any $n\geq 1$ there is exactly one solution to $n=a+b$ with $a\in A$ and $b\in B$.

This was formalized in Lean by van Doorn using Aristotle.
-/
@[category research solved, AMS 11, formal_proof using lean4 at "https://github.com/Woett/Lean-files/blob/d30552f64c55686d40b928a0a3b8e2396357a4ee/ErdosProblem331.lean"]
theorem erdos_331 :
    answer(False) ↔
      ∀ A B : Set ℕ,
      (fun (n : ℕ) ↦ (n : ℝ) ^ (1 / 2 : ℝ)) =O[atTop] (fun (n : ℕ) ↦ (count (· ∈ A) n : ℝ)) →
      (fun (n : ℕ) ↦ (n : ℝ) ^ (1 / 2 : ℝ)) =O[atTop] (fun (n : ℕ) ↦ (count (· ∈ B) n : ℝ)) →
      { s : ℕ × ℕ × ℕ × ℕ | let ⟨a₁, a₂, b₁, b₂⟩ := s
        a₁ ∈ A ∧ a₂ ∈ A ∧ b₁ ∈ B ∧ b₂ ∈ B ∧
        a₁ ≠ a₂ ∧ a₁ + b₂ = a₂ + b₁ }.Infinite := by
  sorry

open scoped Classical in
/--
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 $B$.
-/
@[category research open, AMS 11]
theorem erdos_331.variants.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⟩ := s
        a₁ ∈ A ∧ a₂ ∈ A ∧ b₁ ∈ B ∧ b₂ ∈ B ∧
        a₁ ≠ a₂ ∧ a₁ + b₂ = a₂ + b₁ }.Infinite := by
  sorry
end Erdos331

• Formalization source ↗

On-site verification needs an executable Lean target. None is attached yet.

A source attachment or link is not a verified proof. The statement’s correspondence with the original question needs human review.