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import FormalConjecturesUtilErdős Problem 298
References:
[Bl21] Bloom, T. F., On a density conjecture about unit fractions. arXiv:2112.03726 (2021).
namespace Erdos298Does every set $A \subseteq \mathbb{N}$ of positive density contain some finite $S \subset A$ such that $\sum_{n \in S} \frac{1}{n} = 1$?
The answer is yes, proved by Bloom [Bl21] (even if 'positive density' is interpreted as 'positive upper density', which is likely what Erdős intended).
The theorem below uses the positive-upper-density interpretation; the literal natural-density interpretation is recorded separately.
This was formalized in Lean 3 by Bloom and Mehta.
@[category research solved, AMS 11, formal_proof using other_system at "https://github.com/b-mehta/unit-fractions/blob/master/src/final_results.lean"]
theorem erdos_298 : answer(True) ↔ (∀ (A : Set ℕ), 0 ∉ A → 0 < A.upperDensity →
∃ (S : Finset ℕ), ↑S ⊆ A ∧ ∑ n ∈ S, (1 / n : ℚ) = 1) := ⊢ True ↔ ∀ (A : Set ℕ), 0 ∉ A → 0 < A.upperDensity → ∃ S, ↑S ⊆ A ∧ ∑ n ∈ S, 1 / ↑n = 1
All goals completed! 🐙The literal natural-density interpretation of Erdős Problem 298 follows from [Bl21].
@[category research solved, AMS 11]
theorem erdos_298.variants.natural_density : answer(True) ↔ (∀ (A : Set ℕ), 0 ∉ A →
A.HasPosDensity →
∃ (S : Finset ℕ), ↑S ⊆ A ∧ ∑ n ∈ S, (1 / n : ℚ) = 1) := ⊢ True ↔ ∀ (A : Set ℕ), 0 ∉ A → A.HasPosDensity → ∃ S, ↑S ⊆ A ∧ ∑ n ∈ S, 1 / ↑n = 1
All goals completed! 🐙end Erdos298