1. FormalConjectures
  2. ErdosProblems
  3. «317»
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Erdős Problem 317

Reference: erdosproblems.com/317

namespace Erdos317open Finsetopen Filter

Is there some constant $c>0$ such that for every $n\geq 1$ there exists some $\delta_k\in {-1,0,1}$ for $1\leq k\leq n$ with $$0< \left\lvert \sum_{1\leq k\leq n}\frac{\delta_k}{k}\right\rvert < \frac{c}{2^n}?$$

@[category research open, AMS 11] theorem declaration uses 'sorry'erdos_317 : answer(sorry) ↔ ∃ c > 0, ∀ n ≥ 1, ∃ δ : Fin n → ℚ, Set.range δ ⊆ {-1, 0, 1} ∧ letI lhs : ℝ := |∑ k, (δ k) / (k + 1)| 0 < lhs ∧ lhs < c / 2^n := ⊢ True ↔ ∃ c > 0, ∀ n ≥ 1, ∃ δ, Set.range δ ⊆ {-1, 0, 1} ∧ 0 < |∑ k, ↑(δ k) / (↑↑k + 1)| ∧ |∑ k, ↑(δ k) / (↑↑k + 1)| < c / 2 ^ n All goals completed! 🐙

Is it true that for sufficiently large $n$, for any $\delta_k\in {-1,0,1}$, $$\left\lvert \sum_{1\leq k\leq n}\frac{\delta_k}{k}\right\rvert > \frac{1}{[1,\ldots,n]}$$ whenever the left-hand side is not zero?

@[category research open, AMS 11] theorem declaration uses 'sorry'erdos_317.variants.claim2 : answer(sorry) ↔ ∀ᶠ n in atTop, ∀ δ : (Fin n) → ℚ, δ '' Set.univ ⊆ {-1,0,1} → letI lhs := |∑ k, ((δ k : ℚ) / (k + 1))| lhs ≠ 0 → lhs > 1 / (Icc 1 n).lcm id := ⊢ True ↔ ∀ᶠ (n : ℕ) in atTop, ∀ (δ : Fin n → ℚ), δ '' Set.univ ⊆ {-1, 0, 1} → |∑ k, δ k / (↑↑k + 1)| ≠ 0 → |∑ k, δ k / (↑↑k + 1)| > 1 / ↑((Icc 1 n).lcm id) All goals completed! 🐙

Inequality in erdos_317.variants.claim2 is obvious, the problem is strict inequality.

@[category textbook, AMS 11] lemma declaration uses 'sorry'claim2_inequality : ∀ᶠ n in atTop, ∀ δ : (Fin n) → ℚ, δ '' Set.univ ⊆ {-1,0,1} → letI lhs := |∑ k, ((δ k : ℚ) / (k + 1))| lhs ≠ 0 → lhs ≥ 1 / (Icc 1 n).lcm id := ⊢ ∀ᶠ (n : ℕ) in atTop, ∀ (δ : Fin n → ℚ), δ '' Set.univ ⊆ {-1, 0, 1} → |∑ k, δ k / (↑↑k + 1)| ≠ 0 → |∑ k, δ k / (↑↑k + 1)| ≥ 1 / ↑((Icc 1 n).lcm id) All goals completed! 🐙

erdos_317.variants.claim2 fails for small $n$, for example $$\frac{1}{2}-\frac{1}{3}-\frac{1}{4}=-\frac{1}{12}.$$

@[category textbook, AMS 11] theorem erdos_317.variants.counterexample : ¬ (∀ δ : (Fin 4) → ℚ, δ '' Set.univ ⊆ {-1,0,1} → letI lhs := |∑ k, ((δ k : ℚ) / (k + 1))| lhs ≠ 0 → lhs > (1 : ℚ) / ((Icc 1 4).lcm id : ℕ)) := ⊢ ¬∀ (δ : Fin 4 → ℚ), δ '' Set.univ ⊆ {-1, 0, 1} → |∑ k, δ k / (↑↑k + 1)| ≠ 0 → |∑ k, δ k / (↑↑k + 1)| > 1 / ↑((Icc 1 4).lcm id) ⊢ ∃ δ, δ '' Set.univ ⊆ {-1, 0, 1} ∧ |∑ k, δ k / (↑↑k + 1)| ≠ 0 ∧ |∑ k, δ k / (↑↑k + 1)| ≤ 1 / ↑((Icc 1 4).lcm id) ⊢ ![0, 1, -1, -1] '' Set.univ ⊆ {-1, 0, 1} ∧ |∑ k, ![0, 1, -1, -1] k / (↑↑k + 1)| ≠ 0 ∧ |∑ k, ![0, 1, -1, -1] k / (↑↑k + 1)| ≤ 1 / ↑((Icc 1 4).lcm id) ⊢ {1, 0} ⊆ {0, 1} ∧ 1 / 12 ≤ (↑((Icc 1 4).lcm id))⁻¹ exact ⟨⊢ {1, 0} ⊆ {0, 1} All goals completed! 🐙, ⊢ 1 / 12 ≤ (↑((Icc 1 4).lcm id))⁻¹ ⊢ 12⁻¹ ≤ (↑((Icc 1 4).lcm id))⁻¹; All goals completed! 🐙⟩ end Erdos317