FormalConjectures.ErdosProblems.«332»

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import FormalConjecturesUtil

Erdős Problem 332

Reference: erdosproblems.com/332


namespace Erdos332


open Set

The set of numbers $D(A)$ which occur infinitely often as $a_1 - a_2$ with $a_1, a_2 \in A$.

noncomputable def D_A (A : Set ℕ) : Set ℤ :=
  { d : ℤ | Set.Infinite { (a, b) : ℕ × ℕ | a ∈ A ∧ b ∈ A ∧ (a : ℤ) - (b : ℤ) = d } }

A set $S \subseteq \mathbb{Z}$ has bounded gaps if it is syndetic, meaning there is a uniform bound $M$ such that every interval of length $M$ contains an element of $S$.

def HasBoundedGaps (S : Set ℤ) : Prop :=
  ∃ M : ℕ, M > 0 ∧ ∀ z : ℤ, ∃ s ∈ S, z ≤ s ∧ s < z + (M : ℤ)

Let $A\subseteq \mathbb{N}$ and $D(A)$ be the set of those numbers which occur infinitely often as $a_1 - a_2$ with $a_1, a_2\in A$. What conditions on $A$ are sufficient to ensure $D(A)$ has bounded gaps?

This is formalised here using the answer(sorry) mechanism. In order to solve this problem one has to provide what the sufficient conditions are, and proof that they imply the desired condition. If the condition is a solution to the problem is up to human judgement.

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_332 (A : Set ℕ) : (answer(sorry) : Set ℕ → Prop) A → HasBoundedGaps (D_A A) := byA:Set ℕ⊢ sorry A → HasBoundedGaps (D_A A)
  sorryAll goals completed! 🐙


-- TODO(firsching): formalize additional statements

end Erdos332