File size: 3,431 Bytes
f1f7953 | 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 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 | import OAI.Analysis.IrrationalRotation.Sampling.PrescribedPhaseRealization
import OAI.Analysis.IrrationalRotation.Sampling.CharacterLawConvergence
import Mathlib.Analysis.SpecificLimits.Basic
namespace OAI
namespace Invariant233
namespace PrescribedPhaseRealization
open Filter
open scoped Topology
theorem exists_phase_approximation_family {θ : ℝ} (hθ : Irrational θ)
(target : ℕ → ℝ) (htarget₀ : ∀ i, 0 < target i)
(htarget₁ : ∀ i, target i < 1) {q₀ : ℕ} (hq₀ : 0 < q₀) :
∃ q : ℕ → ℕ → ℕ,
(∀ j, q j 0 = q₀) ∧ (∀ j i, 0 < q j i) ∧
(∀ j, StrictMono (q j)) ∧ (∀ j i, q j i ∣ q j (i + 1)) ∧
(∀ i, Tendsto (fun j => (q j (i + 1) : ℝ) / (q j i : ℝ)) atTop atTop) ∧
∀ i, Tendsto (fun j => Int.fract ((q j (i + 1) : ℝ) * θ)) atTop
(𝓝 (target i)) := by
classical
have hex : ∀ j : ℕ, ∃ q : ℕ → ℕ,
q 0 = q₀ ∧ (∀ i, 0 < q i) ∧
∀ i, q i ∣ q (i + 1) ∧ q i < q (i + 1) ∧
(j + 1) * q i ≤ q (i + 1) ∧
|Int.fract ((q (i + 1) : ℝ) * θ) - target i| < 1 / ((j : ℝ) + 1) := by
intro j
exact exists_divisible_phase_approximation hθ target
(fun _ => 1 / ((j : ℝ) + 1)) (fun _ => j + 1)
htarget₀ htarget₁ (fun _ => by positivity) hq₀
choose q hzero hpos hstep using hex
refine ⟨q, hzero, hpos, ?_, ?_, ?_, ?_⟩
· intro j
exact strictMono_nat_of_lt_succ (fun i => (hstep j i).2.1)
· intro j i
exact (hstep j i).1
· intro i
apply tendsto_atTop_mono (f := fun j : ℕ => (j : ℝ) + 1)
· intro j
apply (le_div_iff₀ (by exact_mod_cast hpos j i)).mpr
exact_mod_cast (hstep j i).2.2.1
· exact tendsto_atTop_add_const_right _ 1 tendsto_natCast_atTop_atTop
· intro i
apply tendsto_iff_dist_tendsto_zero.mpr
apply squeeze_zero (fun _ => dist_nonneg)
(fun j => ?_) tendsto_one_div_add_atTop_nhds_zero_nat
simpa only [Real.dist_eq] using (hstep j i).2.2.2.le
theorem exists_joint_phase_character_limit {θ : ℝ} (hθ : Irrational θ)
(target : ℕ → ℝ) (htarget₀ : ∀ i, 0 < target i)
(htarget₁ : ∀ i, target i < 1) {q₀ : ℕ} (hq₀ : 0 < q₀) :
∃ q : ℕ → ℕ → ℕ,
(∀ j, q j 0 = q₀) ∧ (∀ j i, 0 < q j i) ∧
(∀ j, StrictMono (q j)) ∧ (∀ j i, q j i ∣ q j (i + 1)) ∧
(∀ i, Tendsto (fun j => (q j (i + 1) : ℝ) / (q j i : ℝ)) atTop atTop) ∧
(∀ i, Tendsto (fun j => Int.fract ((q j (i + 1) : ℝ) * θ)) atTop
(𝓝 (target i))) ∧
∀ m : ℕ, Tendsto (fun j => CharacterLaws.characterPhaseLaw
(fun i : Fin m => q j (i.val + 1))
(fun i : Fin m => Int.fract ((q j (i.val + 1) : ℝ) * θ))) atTop
(𝓝 (CharacterLaws.haarPhaseLaw m (fun i : Fin m => target i.val))) := by
obtain ⟨q, hzero, hpos, hstrict, hdiv, hgap, hphase⟩ :=
exists_phase_approximation_family hθ target htarget₀ htarget₁ hq₀
refine ⟨q, hzero, hpos, hstrict, hdiv, hgap, hphase, ?_⟩
intro m
apply CharacterLaws.tendsto_characterPhaseLaw (fun j i => q j (i + 1)) m
· intro j i
exact hpos j (i + 1)
· intro j i k hik
exact (hstrict j).monotone (Nat.add_le_add_right hik 1)
· intro i _hi
exact hgap (i + 1)
· exact tendsto_pi_nhds.mpr (fun i : Fin m => hphase i.val)
end PrescribedPhaseRealization
end Invariant233
end OAI
|