FormalConjectures.ErdosProblems.«264»

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import FormalConjecturesUtil

Erdős Problem 264

Reference: erdosproblems.com/264


namespace Erdos264


open Filter


open scoped ENNReal Asymptotics

A sequence $a_n$ of integers is called an irrationality sequence if for every bounded sequence of integers $b_n$ with $a_n + b_n \neq 0$ and $b_n \neq 0$ for all $n$, the sum $$ \sum \frac{1}{a_n + b_n} $$ is irrational.

Note: there are other possible definitions of this concept. See FormalConjectures/ErdosProblems/263.lean for another possible definition.

def IsIrrationalitySequence (a : ℕ → ℕ) : Prop := ∀ b : ℕ → ℤ,
  BddAbove (Set.range b) → BddBelow (Set.range b) →
  0 ∉ Set.range (fun n ↦ (a n : ℤ) + b n) → 0 ∉ Set.range b →
  Irrational (∑' n, (1 : ℝ) / ((a n : ℤ) + b n))

Is $2^n$ an example of an irrationality sequence? Kovač and Tao proved that it is not [KoTa24]

[KoTa24] Kovač, V. and Tao T., On several irrationality problems for Ahmes series. arXiv:2406.17593 (2024).

@[category research solved, AMS 11]
theorem declaration uses 'sorry'erdos_264.parts.i : ¬IsIrrationalitySequence (2 ^ ·) := by⊢ ¬IsIrrationalitySequence fun x => 2 ^ x sorryAll goals completed! 🐙

Is $n!$ an example of an irrationality sequence?

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_264.parts.ii : answer(sorry) ↔ IsIrrationalitySequence Nat.factorial := by⊢ True ↔ IsIrrationalitySequence Nat.factorial sorryAll goals completed! 🐙

One example is $2^{2^n}$.

@[category research solved, AMS 11]
theorem declaration uses 'sorry'erdos_264.variants.example : IsIrrationalitySequence (fun n ↦ 2 ^ (2 ^ n)) := by⊢ IsIrrationalitySequence fun n => 2 ^ 2 ^ n sorryAll goals completed! 🐙

Kovač and Tao [KoTa24] generally proved that any strictly increasing sequence of positive integers $a_n$ such that $\sum \frac{1}{a_n}$ converges and $$ \liminf_{n \to \infty} (a_n^2 \sum_{k > n} \frac{1}{a_k^2}) > 0 $$ is not an irrationality sequence.

[KoTa24] Kovač, V. and Tao T., On several irrationality problems for Ahmes series. arXiv:2406.17593 (2024).

@[category research solved, AMS 11]
theorem declaration uses 'sorry'erdos_264.variants.ko_tao_neg {a : ℕ → ℕ} (h₁ : StrictMono a) (h₂ : 0 ∉ Set.range a)
    (h₃ : Summable ((1 : ℝ) / a ·))
    (h₄ : 0 < atTop.liminf fun n ↦ a n ^ 2 * ∑' k : Set.Ioi n, (1 : ℝ) / a k ^ 2) :
    ¬IsIrrationalitySequence a := bya:ℕ → ℕh₁:StrictMono ah₂:0 ∉ Set.range ah₃:Summable fun x => 1 / ↑(a x)h₄:0 < liminf (fun n => ↑(a n) ^ 2 * ∑' (k : ↑(Set.Ioi n)), 1 / ↑(a ↑k) ^ 2) atTop⊢ ¬IsIrrationalitySequence a
  sorryAll goals completed! 🐙

On the other hand, Kovač and Tao [KoTa24] do prove that for any function $F$ with $\lim_{n \to \infty} \frac{F(n + 1)}{F(n)} = \infty$ there exists such an irrationality sequence with $a_n \sim F(n)$.

[KoTa24] Kovač, V. and Tao T., On several irrationality problems for Ahmes series. arXiv:2406.17593 (2024).

@[category research solved, AMS 11]
theorem declaration uses 'sorry'erdos_264.variants.ko_tao_pos {F : ℕ → ℕ}
    (hF : atTop.Tendsto (fun n ↦ (F (n + 1) : ℝ) / F n) atTop) :
    ∃ a : ℕ → ℕ, IsIrrationalitySequence a ∧ (fun n ↦ (a n : ℝ)) ~[atTop] fun n ↦ (F n : ℝ) := byF:ℕ → ℕhF:Tendsto (fun n => ↑(F (n + 1)) / ↑(F n)) atTop atTop⊢ ∃ a, IsIrrationalitySequence a ∧ (fun n => ↑(a n)) ~[atTop] fun n => ↑(F n)
  sorryAll goals completed! 🐙


end Erdos264