| import OAI.NumberTheory.PiExponent.Analysis.DyadicSpacing |
| import OAI.NumberTheory.PiExponent.Analysis.FlintSine |
| import OAI.NumberTheory.PiExponent.Statement |
|
|
| namespace OAI |
|
|
| namespace PiExponent |
|
|
| theorem uniform_pi_distance_of_eventual_lower_bound (h : PiEventualLowerBound) |
| {ν : ℝ} (hν : 2 < ν) : |
| ∃ c : ℝ, 0 < c ∧ c ≤ 1 ∧ ∀ q : ℕ, 0 < q → |
| c * (q : ℝ) ^ (1 - ν) ≤ integerDistance (q * Real.pi) := by |
| obtain ⟨Q, hQ, hbound⟩ := h ν hν |
| apply uniform_integerDistance_lower_bound irrational_pi |
| exact ⟨Q, fun q hq _ p => hbound p q hq⟩ |
|
|
| theorem flintHills_summable_of_eventual_lower_bound (h : PiEventualLowerBound) : |
| Summable (fun n : ℕ => 1 / ((n : ℝ) ^ 3 * Real.sin n ^ 2)) := by |
| obtain ⟨c, hc, _, hbound⟩ := |
| uniform_pi_distance_of_eventual_lower_bound h (ν := 9 / 4) (by norm_num) |
| exact flintHills_summable_of_distance_summable |
| (summable_inverse_cube_integerDistance hc (by norm_num) (by norm_num) hbound) |
|
|
| theorem flintHills_positive_summable_of_eventual_lower_bound (h : PiEventualLowerBound) : |
| Summable (fun n : ℕ => |
| 1 / (((n + 1 : ℕ) : ℝ) ^ 3 * Real.sin (n + 1 : ℕ) ^ 2)) := |
| (summable_nat_add_iff 1).mpr (flintHills_summable_of_eventual_lower_bound h) |
|
|
| end PiExponent |
|
|
| end OAI |
|
|
