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import FormalConjecturesUtilErdős Problem 299
[Bl21] Bloom, T. F., On a density conjecture about unit fractions. arXiv:2112.03726 (2021).
open Filter
namespace Erdos299
Is there an infinite sequence $a_1 < a_2 < \dots$ such that $a_{i+1} - a_i = O(1)$ and no finite sum of $\frac{1}{a_i}$ is equal to 1?
There does not exist such a sequence, which follows from the positive solution to [erdosproblems.com/298] by Bloom [Bl21].
This was formalized in Lean 3 by Bloom and Mehta.
@[category research solved, AMS 11 40, formal_proof using other_system at "https://github.com/b-mehta/unit-fractions/blob/master/src/final_results.lean"]
theorem erdos_299 : answer(False) ↔ (∃ (a : ℕ → ℕ),
StrictMono a ∧ (∀ n, 0 < a n) ∧
(fun n ↦ (a (n + 1) : ℝ) - a n) =O[atTop] (1 : ℕ → ℝ) ∧
∀ S : Finset ℕ, ∑ i ∈ S, (1 : ℝ) / a i ≠ 1) := ⊢ False ↔
∃ a,
StrictMono a ∧
(∀ (n : ℕ), 0 < a n) ∧ (fun n => ↑(a (n + 1)) - ↑(a n)) =O[atTop] 1 ∧ ∀ (S : Finset ℕ), ∑ i ∈ S, 1 / ↑(a i) ≠ 1
All goals completed! 🐙
The corresponding question is also false if one replaces sequences such that $a_{i+1} - a_i = O(1)$ with sets of positive density, as follows from [Bl21].
The statement is as follows: If $A \subset \mathbb{N}$ has positive upper density (and hence certainly if $A$ has positive density) then there is a finite $S \subset A$ such that $\sum_{n \in S} \frac{1}{n} = 1$.
@[category research solved, AMS 11 40]
theorem erdos_299.variants.density : ∀ (A : Set ℕ), 0 ∉ A → 0 < A.upperDensity →
∃ S : Finset ℕ, ↑S ⊆ A ∧ ∑ n ∈ S, (1 : ℝ) / n = 1 := ⊢ ∀ (A : Set ℕ), 0 ∉ A → 0 < A.upperDensity → ∃ S, ↑S ⊆ A ∧ ∑ n ∈ S, 1 / ↑n = 1
All goals completed! 🐙
end Erdos299