Erdős Problem 873

Reference: erdosproblems.com/873

@[reducible, inline]

noncomputable abbrev Erdos873.F (a : ℕ → ℕ) (X : ℝ) (k : ℕ) : ℕ∞

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.

Equations

• Erdos873.F a X k = {i : ℕ | ↑((Finset.range k).lcm fun (m : ℕ) => a (i + m)) < X}.encard

theorem Erdos873.erdos_873 : (∀ (a : ℕ → ℕ), ∀ ε > 0, 0 < a 0 → StrictMono a → ∃ (k : ℕ), ∀ X > 0, ↑↑(F a X k) < ↑(X ^ ε)) ↔ sorry

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 $F(A,X,k) < X^\epsilon$?