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b797a5b | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 | 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
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