FormalConjectures.ErdosProblems.«873»

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import FormalConjecturesUtil

Erdős Problem 873

Reference: erdosproblems.com/873


namespace Erdos873

Let $a$ be some sequence of natural numbers. We set $F(A,X,k)$ to be the count of the number of $i$ such that $[a_i,a_{i+1}, \dots ,a_{i+k−1}] < X$, where the left-hand side is the least common multiple.

noncomputable abbrev F (a : ℕ → ℕ) (X : ℝ) (k : ℕ) : ℕ∞ :=
  {i : ℕ | (Finset.range k).lcm (fun m => a (i + m)) < X}.encard

Let $A = {a_1 < a_2 < \dots} \subseteq \mathbb{N}$ and let $F(A,X,k)$ count the number of $i$ such that $[a_i,a_{i+1}, \dots ,a_{i+k−1}] < X$, where the left-hand side is the least common multiple. Is it true that, for every $\epsilon > 0$, there exists some $k$ such that $F(A,X,k) < X^\epsilon$?

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_873 : answer(sorry) ↔ ∀ᵉ (a : ℕ → ℕ) (ε > (0 : ℝ)), 0 < a 0 → StrictMono a →
    ∃ k, ∀ X > 0, F a X k < (X^ε).toEReal := by⊢ True ↔ ∀ (a : ℕ → ℕ), ∀ ε > 0, 0 < a 0 → StrictMono a → ∃ k, ∀ X > 0, ↑↑(F a X k) < ↑(X ^ ε)
  sorryAll goals completed! 🐙


end Erdos873