| import OAI.Analysis.FiniteFactor.Polynomial |
| import Mathlib.MeasureTheory.Integral.Bochner.Set |
| import Mathlib.Analysis.SpecificLimits.Basic |
| import Mathlib.Analysis.SpecialFunctions.Exp |
| import Mathlib.Tactic.Linarith |
| import Mathlib.Tactic.Ring |
|
|
| namespace OAI |
|
|
| namespace FiniteFactor |
| open MeasureTheory Filter |
| open scoped Topology |
|
|
| / |
| noncomputable def polynomialEpsilon (K : ℕ) (D δ : ℝ) (hδ : 0 < δ) : ℝ := |
| (laurent_product_uniform_lower_bound K D δ hδ).choose |
|
|
| theorem polynomialEpsilon_pos (K : ℕ) (D δ : ℝ) (hδ : 0 < δ) : |
| 0 < polynomialEpsilon K D δ hδ := |
| (laurent_product_uniform_lower_bound K D δ hδ).choose_spec.1 |
|
|
| noncomputable def coefficientMass (K : ℕ) (A B : ℤ → ℂ) : ℝ := |
| ∑ n ∈ Finset.range (2*K+1), ‖laurentProductCoefficient K A B (-(n : ℤ))‖ |
|
|
| theorem polynomialEpsilon_le_mass (K : ℕ) (D δ : ℝ) (hδ : 0 < δ) |
| (A B : ℤ → ℂ) |
| (hA : ∀ v : ℤ, -(K : ℤ) ≤ v → v ≤ K → ‖A v‖ ≤ D) |
| (hB : ∀ v : ℤ, -(K : ℤ) ≤ v → v ≤ K → ‖B v‖ ≤ D) |
| (hA0 : δ ≤ ‖A 0‖) (hB0 : δ ≤ ‖B 0‖) : |
| polynomialEpsilon K D δ hδ ≤ coefficientMass K A B := by |
| obtain ⟨n, hn, hbound⟩ := |
| (laurent_product_uniform_lower_bound K D δ hδ).choose_spec.2 A B hA hB hA0 hB0 |
| have hnmem : n ∈ Finset.range (2*K+1) := Finset.mem_range.mpr (by omega) |
| apply hbound.trans |
| unfold coefficientMass |
| exact Finset.single_le_sum (fun (i : ℕ) _ => norm_nonneg (laurentProductCoefficient K A B (-(i : ℤ)))) hnmem |
|
|
| variable {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [IsFiniteMeasure μ] |
|
|
| / |
| event, using only almost-everywhere coefficient bounds. -/ |
| theorem polynomial_event_integral_lower (K : ℕ) (D δ : ℝ) (hδ : 0 < δ) |
| (A B : Ω → ℤ → ℂ) {E : Set Ω} (hE : MeasurableSet E) |
| (hint : IntegrableOn (fun ω => coefficientMass K (A ω) (B ω)) E μ) |
| (hA : ∀ᵐ ω ∂μ, ω ∈ E → |
| ∀ v : ℤ, -(K : ℤ) ≤ v → v ≤ K → ‖A ω v‖ ≤ D) |
| (hB : ∀ᵐ ω ∂μ, ω ∈ E → |
| ∀ v : ℤ, -(K : ℤ) ≤ v → v ≤ K → ‖B ω v‖ ≤ D) |
| (hA0 : ∀ᵐ ω ∂μ, ω ∈ E → δ ≤ ‖A ω 0‖) |
| (hB0 : ∀ᵐ ω ∂μ, ω ∈ E → δ ≤ ‖B ω 0‖) : |
| polynomialEpsilon K D δ hδ * μ.real E ≤ |
| ∫ ω in E, coefficientMass K (A ω) (B ω) ∂μ := by |
| have hlower : ∀ᵐ ω ∂μ, ω ∈ E → |
| polynomialEpsilon K D δ hδ ≤ coefficientMass K (A ω) (B ω) := by |
| filter_upwards [hA, hB, hA0, hB0] with ω ha hb ha0 hb0 hω |
| exact polynomialEpsilon_le_mass K D δ hδ _ _ (ha hω) (hb hω) (ha0 hω) (hb0 hω) |
| have hi := setIntegral_mono_on_ae |
| (integrable_const (polynomialEpsilon K D δ hδ)) hint hE hlower |
| rw [setIntegral_const, smul_eq_mul, mul_comm] at hi |
| exact hi |
|
|
| omit [IsFiniteMeasure μ] in |
| / |
| theorem coefficientMass_integral_upper (K : ℕ) (A B : Ω → ℤ → ℂ) |
| {E : Set Ω} {a : ℝ} |
| (hint : ∀ n ≤ 2*K, IntegrableOn |
| (fun ω => ‖laurentProductCoefficient K (A ω) (B ω) (-(n : ℤ))‖) E μ) |
| (hupper : ∀ n ≤ 2*K, |
| (∫ ω in E, ‖laurentProductCoefficient K (A ω) (B ω) (-(n : ℤ))‖ ∂μ) ≤ a) : |
| (∫ ω in E, coefficientMass K (A ω) (B ω) ∂μ) ≤ (2*(K : ℝ)+1)*a := by |
| unfold coefficientMass |
| rw [integral_finsetSum (Finset.range (2*K+1)) (fun n hn => hint n (by simpa using Nat.le_of_lt_succ (Finset.mem_range.mp hn)))] |
| calc |
| _ ≤ ∑ n ∈ Finset.range (2*K+1), a := |
| Finset.sum_le_sum (fun n hn => hupper n (Nat.le_of_lt_succ (Finset.mem_range.mp hn))) |
| _ = _ := by simp [Nat.cast_add, Nat.cast_mul] |
|
|
| / |
| theorem exponential_tail_relaxation {L : ℝ} (hL : 0 ≤ L) : |
| Real.exp (-L) + 2 * Real.exp (-L/800) ≤ 3 * Real.exp (-L/10000) := by |
| have h₁ : Real.exp (-L) ≤ Real.exp (-L/10000) := Real.exp_le_exp.mpr (by linarith) |
| have h₂ : Real.exp (-L/800) ≤ Real.exp (-L/10000) := Real.exp_le_exp.mpr (by linarith) |
| linarith |
|
|
| theorem no_eventual_small_convolutions |
| {Ωs : ℕ → Type*} [∀ j, MeasurableSpace (Ωs j)] |
| (μs : ∀ j, Measure (Ωs j)) [∀ j, IsFiniteMeasure (μs j)] |
| (K : ℕ → ℕ) (D δ : ℕ → ℝ) (hδ : ∀ j, 0 < δ j) |
| (A B : ∀ j, Ωs j → ℤ → ℂ) (E : ∀ j, Set (Ωs j)) |
| (hE : ∀ j, MeasurableSet (E j)) |
| (hint : ∀ j, IntegrableOn (fun ω => coefficientMass (K j) (A j ω) (B j ω)) (E j) (μs j)) |
| (hA : ∀ j, ∀ᵐ ω ∂(μs j), ω ∈ E j → |
| ∀ v : ℤ, -(K j : ℤ) ≤ v → v ≤ K j → ‖A j ω v‖ ≤ D j) |
| (hB : ∀ j, ∀ᵐ ω ∂(μs j), ω ∈ E j → |
| ∀ v : ℤ, -(K j : ℤ) ≤ v → v ≤ K j → ‖B j ω v‖ ≤ D j) |
| (hA0 : ∀ j, ∀ᵐ ω ∂(μs j), ω ∈ E j → δ j ≤ ‖A j ω 0‖) |
| (hB0 : ∀ j, ∀ᵐ ω ∂(μs j), ω ∈ E j → δ j ≤ ‖B j ω 0‖) |
| {c : ℝ} (hc : 0 < c) |
| (hmass : ∀ᶠ j in atTop, c ≤ (μs j).real (E j)) |
| (hupper : ∀ᶠ j in atTop, |
| (∫ ω in E j, coefficientMass (K j) (A j ω) (B j ω) ∂(μs j)) ≤ |
| polynomialEpsilon (K j) (D j) (δ j) (hδ j) / ((j : ℝ)+1)) : False := by |
| have hsmall : ∀ᶠ (j : ℕ) in atTop, c ≤ 1 / ((j : ℝ)+1) := by |
| filter_upwards [hmass, hupper] with j hm hu |
| have hp := polynomialEpsilon_pos (K j) (D j) (δ j) (hδ j) |
| have hl := polynomial_event_integral_lower (K j) (D j) (δ j) (hδ j) |
| (A j) (B j) (hE j) (hint j) (hA j) (hB j) (hA0 j) (hB0 j) |
| have hmul : polynomialEpsilon (K j) (D j) (δ j) (hδ j) * c ≤ |
| polynomialEpsilon (K j) (D j) (δ j) (hδ j) * (1 / ((j : ℝ)+1)) := by |
| calc |
| _ ≤ polynomialEpsilon (K j) (D j) (δ j) (hδ j) * (μs j).real (E j) := |
| mul_le_mul_of_nonneg_left hm hp.le |
| _ ≤ _ := hl |
| _ ≤ polynomialEpsilon (K j) (D j) (δ j) (hδ j) / ((j : ℝ)+1) := hu |
| _ = _ := by ring |
| exact (mul_le_mul_iff_right₀ hp).mp (by simpa only [mul_comm] using hmul) |
| exact (not_le_of_gt hc) (ge_of_tendsto tendsto_one_div_add_atTop_nhds_zero_nat hsmall) |
|
|
| end FiniteFactor |
|
|
| end OAI |
|
|
