| import OAI.NumberTheory.PiExponent.Statement | |
| namespace OAI | |
| namespace PiExponent | |
| theorem eventualLowerBound_iff_integer (x : ℝ) : | |
| EventualLowerBound x ↔ IntegerEventualLowerBound x := by | |
| unfold EventualLowerBound IntegerEventualLowerBound | |
| constructor | |
| · intro h ν hν | |
| obtain ⟨Q, hQ, hb⟩ := h ν hν | |
| refine ⟨Q, by exact_mod_cast hQ, ?_⟩ | |
| intro p q hq | |
| have hq0 : 0 ≤ q := le_trans (by omega) hq | |
| have hnat : Q ≤ q.toNat := by omega | |
| have heq : (q.toNat : ℝ) = (q : ℝ) := by | |
| exact_mod_cast (Int.toNat_of_nonneg hq0) | |
| simpa only [heq] using hb p q.toNat hnat | |
| · intro h ν hν | |
| obtain ⟨Q, hQ, hb⟩ := h ν hν | |
| have hQ0 : 0 ≤ Q := by omega | |
| refine ⟨Q.toNat, by omega, ?_⟩ | |
| intro p q hq | |
| have hqi : Q ≤ (q : ℤ) := by omega | |
| simpa using hb p q hqi | |
| theorem finite_rat_den_le_abs_sub_lt_one (x : ℝ) (N : ℕ) : | |
| {r : ℚ | r.den ≤ N ∧ |x - (r : ℝ)| < 1}.Finite := by | |
| classical | |
| let f : ℚ → ℤ × ℕ := fun r => (r.num, r.den) | |
| have hinj : Function.Injective f := by | |
| intro a b hab | |
| have hp := Prod.mk.inj hab | |
| rw [← Rat.num_div_den a, ← Rat.num_div_den b, hp.1, hp.2] | |
| let t : Set (ℤ × ℕ) := ⋃ (q : ℕ) (_ : q ∈ Set.Icc 1 N), | |
| Set.Icc ⌈(x - 1) * q⌉ ⌊(x + 1) * q⌋ ×ˢ {q} | |
| have ht : t.Finite := | |
| Set.Finite.biUnion (Set.finite_Icc _ _) fun q _ => | |
| Set.Finite.prod (Set.finite_Icc _ _) (Set.finite_singleton _) | |
| have hsub : f '' {r : ℚ | r.den ≤ N ∧ |x - (r : ℝ)| < 1} ⊆ t := by | |
| rintro _ ⟨r, ⟨hrN, hr⟩, rfl⟩ | |
| have hrden : (0 : ℝ) < r.den := Nat.cast_pos.mpr r.pos | |
| have hrlow : x - 1 < (r : ℝ) := by linarith [(abs_lt.mp hr).2] | |
| have hrhigh : (r : ℝ) < x + 1 := by linarith [(abs_lt.mp hr).1] | |
| have hnlow : (x - 1) * r.den ≤ (r.num : ℝ) := by | |
| rw [Rat.cast_def] at hrlow | |
| exact ((lt_div_iff₀ hrden).mp hrlow).le | |
| have hnhigh : (r.num : ℝ) ≤ (x + 1) * r.den := by | |
| rw [Rat.cast_def] at hrhigh | |
| exact ((div_lt_iff₀ hrden).mp hrhigh).le | |
| simp only [t, Set.mem_iUnion] | |
| refine ⟨r.den, ⟨r.pos, hrN⟩, ?_⟩ | |
| exact ⟨⟨Int.ceil_le.mpr hnlow, Int.le_floor.mpr hnhigh⟩, rfl⟩ | |
| exact (ht.subset hsub).of_finite_image hinj.injOn | |
| theorem two_mem_approximationExponents {x : ℝ} (hx : Irrational x) : | |
| (2 : ℝ) ∈ ApproximationExponents x := by | |
| refine ⟨by norm_num, ?_⟩ | |
| have hi := Real.infinite_rat_abs_sub_lt_one_div_den_sq_of_irrational hx | |
| have hf := finite_rat_den_le_abs_sub_lt_one x 1 | |
| apply (hi.sdiff hf).mono | |
| intro r hr | |
| have hrden : 1 ≤ (r.den : ℝ) := by exact_mod_cast r.pos | |
| have hrone : |x - (r : ℝ)| < 1 := | |
| hr.1.trans_le (by | |
| simpa only [div_one] using | |
| one_div_le_one_div_of_le (by norm_num : (0 : ℝ) < 1) | |
| (by nlinarith : (1 : ℝ) ≤ (r.den : ℝ)^2)) | |
| have hr2 : 2 ≤ r.den := by | |
| by_contra h | |
| exact hr.2 ⟨by omega, hrone⟩ | |
| refine ⟨hr2, abs_pos.mpr (sub_ne_zero.mpr (hx.ne_rat r)), ?_⟩ | |
| simpa only [Set.mem_ofPred_eq, Real.rpow_neg (Nat.cast_nonneg r.den), | |
| Real.rpow_two, one_div] using hr.1 | |
| theorem finite_goodRationalApproximations_of_eventualLowerBound {x ν : ℝ} | |
| (hx : EventualLowerBound x) (hν : 2 < ν) : | |
| (GoodRationalApproximations x ν).Finite := by | |
| obtain ⟨Q, hQ, hb⟩ := hx ν hν | |
| apply (finite_rat_den_le_abs_sub_lt_one x Q).subset | |
| intro r hr | |
| have hltQ : r.den < Q := by | |
| by_contra h | |
| have h := hb r.num r.den (by omega) | |
| rw [← Rat.cast_def] at h | |
| exact (not_lt_of_ge h) hr.2.2 | |
| refine ⟨hltQ.le, hr.2.2.trans_le ?_⟩ | |
| exact Real.rpow_le_one_of_one_le_of_nonpos | |
| (by exact_mod_cast r.pos) (by linarith) | |
| theorem irrationalityExponent_eq_two_of_eventualLowerBound {x : ℝ} | |
| (hi : Irrational x) (hx : EventualLowerBound x) : | |
| irrationalityExponent x = 2 := by | |
| have htwo := two_mem_approximationExponents hi | |
| have hbound : ∀ ν ∈ ApproximationExponents x, ν ≤ 2 := by | |
| intro ν hν | |
| by_contra h | |
| exact (finite_goodRationalApproximations_of_eventualLowerBound hx (by linarith)).not_infinite hν.2 | |
| apply le_antisymm | |
| · exact csSup_le ⟨2, htwo⟩ hbound | |
| · exact le_csSup ⟨2, hbound⟩ htwo | |
| theorem pi_irrationalityExponent_eq_two_of_eventualLowerBound | |
| (h : PiEventualLowerBound) : irrationalityExponent Real.pi = 2 := | |
| irrationalityExponent_eq_two_of_eventualLowerBound irrational_pi h | |
| theorem eventualLowerBound_of_not_unbounded_approximations {x : ℝ} | |
| (h : ∀ ν : ℝ, 2 < ν → | |
| ¬ ∀ Q : ℕ, ∃ (p : ℤ) (q : ℕ), Q ≤ q ∧ | |
| |x - (p : ℝ) / (q : ℝ)| ≤ (q : ℝ) ^ (-ν)) : | |
| EventualLowerBound x := by | |
| intro ν hν | |
| have hbad := h ν hν | |
| push Not at hbad | |
| obtain ⟨Q, hQ⟩ := hbad | |
| refine ⟨max Q 2, le_max_right _ _, ?_⟩ | |
| intro p q hq | |
| exact (hQ p q ((le_max_left Q 2).trans hq)).le | |
| end PiExponent | |
| end OAI | |