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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 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 | import OAI.Analysis.IrrationalRotation.Sampling.SeparatedCharacterFrequencies
import Mathlib.Analysis.Fourier.AddCircleMulti
import Mathlib.Topology.MetricSpace.Equicontinuity
namespace OAI
noncomputable section
namespace Invariant233
namespace CharacterWeakConvergence
open MeasureTheory Filter Set
open scoped Topology NNReal
section Integral
variable {X Y : Type*} [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MeasurableSpace X] [BorelSpace X]
(μ : Measure X) [IsProbabilityMeasure μ]
def pullbackIntegral (φ : C(X, Y)) : C(Y, ℂ) →L[ℂ] ℂ :=
let L : C(Y, ℂ) →ₗ[ℂ] ℂ :=
{ toFun := fun f => ∫ x, f (φ x) ∂μ
map_add' := by
intro f g
exact integral_add
(f.continuous.comp φ.continuous |>.integrable_of_hasCompactSupport
(HasCompactSupport.of_compactSpace _))
(g.continuous.comp φ.continuous |>.integrable_of_hasCompactSupport
(HasCompactSupport.of_compactSpace _))
map_smul' := by
intro c f
simp only [ContinuousMap.smul_apply, integral_smul, RingHom.id_apply] }
L.mkContinuous 1 (fun f => by
have h := norm_integral_le_of_norm_le_const
(μ := μ) (Filter.Eventually.of_forall (fun x => f.norm_coe_le_norm (φ x)))
change ‖∫ x, f (φ x) ∂μ‖ ≤ 1 * ‖f‖
simpa using h)
@[simp] theorem pullbackIntegral_apply (φ : C(X, Y)) (f : C(Y, ℂ)) :
pullbackIntegral μ φ f = ∫ x, f (φ x) ∂μ := rfl
theorem pullbackIntegral_norm_le (φ : C(X, Y)) : ‖pullbackIntegral μ φ‖ ≤ 1 := by
apply ContinuousLinearMap.opNorm_le_bound _ zero_le_one
intro f
have h := norm_integral_le_of_norm_le_const
(μ := μ) (Filter.Eventually.of_forall (fun x => f.norm_coe_le_norm (φ x)))
simpa using h
end Integral
theorem tendsto_of_dense_span
{E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E]
(L : ℕ → E →L[ℂ] ℂ) (M : E →L[ℂ] ℂ) (C : ℝ≥0)
(hL : ∀ n, ‖L n‖₊ ≤ C) (s : Set E)
(hs : (Submodule.span ℂ s).topologicalClosure = ⊤)
(hconv : ∀ f ∈ s, Tendsto (fun n => L n f) atTop (𝓝 (M f)))
(f : E) : Tendsto (fun n => L n f) atTop (𝓝 (M f)) := by
let S : Submodule ℂ E :=
{ carrier := {f | Tendsto (fun n => L n f) atTop (𝓝 (M f))}
zero_mem' := by simp
add_mem' := by
intro f g hf hg
simpa only [Set.mem_ofPred_eq, map_add] using hf.add hg
smul_mem' := by
intro c f hf
simpa only [Set.mem_ofPred_eq, map_smul] using hf.const_smul c }
have hclosed : IsClosed (S : Set E) := by
apply Equicontinuous.isClosed_setOfPred_tendsto _ M.continuous
exact (LipschitzWith.uniformEquicontinuous (fun n => (L n : E → ℂ)) C
(fun n => (L n).lipschitzWith.weaken (hL n))).equicontinuous
have hsS : Submodule.span ℂ s ≤ S := Submodule.span_le.mpr hconv
have htop : (⊤ : Submodule ℂ E) ≤ S := by
rw [← hs]
exact Submodule.topologicalClosure_minimal _ hsS hclosed
exact htop (Submodule.mem_top)
local instance circleVolumeProbability :
IsProbabilityMeasure (volume : Measure (AddCircle (1 : ℝ))) :=
⟨by simp [AddCircle.measure_univ]⟩
theorem integral_mFourier_eq_zero {m : ℕ} {v : Fin m → ℤ} (hv : v ≠ 0) :
(∫ y : UnitAddTorus (Fin m), UnitAddTorus.mFourier v y) = 0 := by
change (∫ y : Fin m → AddCircle (1 : ℝ), ∏ i, fourier (v i) (y i)) = 0
rw [integral_fintype_prod_volume_eq_prod]
obtain ⟨i, hi⟩ := Function.ne_iff.mp hv
exact Finset.prod_eq_zero (Finset.mem_univ i)
(SeparatedCharacterFrequencies.integral_fourier_eq_zero hi)
theorem tendsto_integral_joint_characters
(q : ℕ → ℕ → ℕ) (m : ℕ)
(hq : ∀ j i, 0 < q j i) (hmono : ∀ j, Monotone (q j))
(hgap : ∀ i, i + 1 < m → Tendsto
(fun j => (q j (i + 1) : ℝ) / (q j i : ℝ)) atTop atTop)
(f : C(UnitAddTorus (Fin m), ℂ)) :
Tendsto (fun j => ∫ x : AddCircle (1 : ℝ),
f (fun i : Fin m => (q j i.val : ℤ) • x)) atTop
(𝓝 (∫ y : UnitAddTorus (Fin m), f y)) := by
let φ : ℕ → C(AddCircle (1 : ℝ), UnitAddTorus (Fin m)) := fun j =>
⟨fun x i => (q j i.val : ℤ) • x, by fun_prop⟩
let L : ℕ → C(UnitAddTorus (Fin m), ℂ) →L[ℂ] ℂ :=
fun j => pullbackIntegral volume (φ j)
let M : C(UnitAddTorus (Fin m), ℂ) →L[ℂ] ℂ :=
pullbackIntegral volume (ContinuousMap.id _)
have hL : ∀ j, ‖L j‖₊ ≤ (1 : ℝ≥0) := by
intro j
exact_mod_cast pullbackIntegral_norm_le volume (φ j)
have hconv : ∀ g ∈ Set.range (UnitAddTorus.mFourier (d := Fin m)),
Tendsto (fun j => L j g) atTop (𝓝 (M g)) := by
rintro _ ⟨v, rfl⟩
by_cases hv : v = 0
· subst v
simp only [UnitAddTorus.mFourier_zero]
simp [L, M, pullbackIntegral_apply]
· let v' : ℕ → ℤ := fun n => if h : n < m then v ⟨n, h⟩ else 0
have hv' : ∃ i < m, v' i ≠ 0 := by
obtain ⟨i, hi⟩ := Function.ne_iff.mp hv
refine ⟨i.val, i.isLt, ?_⟩
simpa [v'] using hi
have hprod : ∀ j (x : AddCircle (1 : ℝ)),
UnitAddTorus.mFourier v (φ j x) =
∏ i ∈ Finset.range m, fourier (v' i) ((q j i : ℤ) • x) := by
intro j x
change (∏ i : Fin m, fourier (v i) ((q j i.val : ℤ) • x)) = _
simpa [v'] using Fin.prod_univ_eq_prod_range
(fun i => fourier (v' i) ((q j i : ℤ) • x)) m
have hevent : ∀ᶠ j in atTop, L j (UnitAddTorus.mFourier v) = 0 := by
filter_upwards [SeparatedCharacterFrequencies.eventually_integral_product_fourier_eq_zero
v' q m hq hmono hgap hv'] with j hj
change (∫ x : AddCircle (1 : ℝ), UnitAddTorus.mFourier v (φ j x)) = 0
simpa only [hprod] using hj
have hM : M (UnitAddTorus.mFourier v) = 0 := integral_mFourier_eq_zero hv
rw [hM]
exact tendsto_const_nhds.congr' (hevent.mono fun _ h => h.symm)
exact tendsto_of_dense_span L M 1 hL _
UnitAddTorus.span_mFourier_closure_eq_top hconv f
end CharacterWeakConvergence
end Invariant233
end
end OAI
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