| import OAI.NumberTheory.PiExponent.Approximation.AdmissibleMatrixGeometry |
| import OAI.NumberTheory.PiExponent.Approximation.InterpolationConsequence |
|
|
| namespace OAI |
|
|
| namespace PiExponent |
|
|
| theorem pi_eventual_lower_bound : PiEventualLowerBound := |
| piEventualLowerBound_of_globalInterpolation AdmissibleMatrixInterpolation.globalInterpolation |
|
|
| theorem pi_integer_eventual_lower_bound : IntegerEventualLowerBound Real.pi := |
| (eventualLowerBound_iff_integer Real.pi).mp pi_eventual_lower_bound |
|
|
| theorem pi_irrationalityExponent_eq_two : irrationalityExponent Real.pi = 2 := |
| pi_irrationalityExponent_eq_two_of_eventualLowerBound pi_eventual_lower_bound |
|
|
| theorem main : |
| (∀ ν : ℝ, 2 < ν → ∃ Q : ℤ, 2 ≤ Q ∧ |
| ∀ p q : ℤ, Q ≤ q → |
| (q : ℝ) ^ (-ν) ≤ |Real.pi - (p : ℝ) / (q : ℝ)|) ∧ |
| sSup {ν : ℝ | 0 < ν ∧ |
| Set.Infinite {r : ℚ | 2 ≤ r.den ∧ |
| 0 < |Real.pi - (r : ℝ)| ∧ |
| |Real.pi - (r : ℝ)| < (r.den : ℝ) ^ (-ν)}} = 2 := by |
| refine ⟨pi_integer_eventual_lower_bound, ?_⟩ |
| change irrationalityExponent Real.pi = 2 |
| exact pi_irrationalityExponent_eq_two |
|
|
| theorem flint_hills_summable : |
| Summable (fun n : ℕ => |
| 1 / (((n + 1 : ℕ) : ℝ) ^ 3 * Real.sin (n + 1 : ℕ) ^ 2)) := |
| flintHills_positive_summable_of_eventual_lower_bound pi_eventual_lower_bound |
|
|
| end PiExponent |
|
|
| end OAI |
|
|
