| import Mathlib |
| import OAI.Analysis.IrrationalRotation.Model.PhaseFailure |
| import OAI.Analysis.IrrationalRotation.Sampling.PhaseFailureSummability |
| import OAI.Analysis.IrrationalRotation.Sampling.IntegerPhaseWindow |
| import OAI.Analysis.IrrationalRotation.Product.IndependentMissBounds |
| import OAI.Analysis.IrrationalRotation.Product.IndependentAngles |
| import OAI.Analysis.IrrationalRotation.Analysis.ModuloIntervalMeasure |
| import OAI.Analysis.IrrationalRotation.Sampling.PhaseRanges |
| import Mathlib.Probability.Independence.InfinitePi |
|
|
| namespace OAI |
|
|
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal Topology |
|
|
| namespace Invariant233.ProductPhases |
|
|
| theorem not_mem_largePhaseFailure_iff {ι : Type*} (J : Finset ι) |
| {N : ℝ} (hN : 0 < N) (θ : ι → ℝ) : |
| θ ∉ largePhaseFailure J N ↔ largePhaseCondition J N θ := by |
| simp only [largePhaseFailure, Set.mem_ofPred_eq, not_exists, not_and, |
| not_lt, largePhaseCondition, mem_largePhaseTimes hN] |
| constructor |
| · intro h k hlo hhi |
| exact h k ⟨hlo, hhi⟩ |
| · intro h k hk |
| exact h k hk.1 hk.2 |
|
|
| theorem measure_largePhaseFailure_le {ι : Type*} |
| (μ : Measure (ι → ℝ)) (J : Finset ι) (N : ℝ) (hm : 100 ≤ (J.card : ℝ)) |
| (hind : ∀ k : ℤ, iIndepSet |
| (fun i => {θ : ι → ℝ | 1 - Int.fract ((k : ℝ) * θ i) ∉ |
| Set.Icc ((J.card : ℝ) ^ (-19 / 20 : ℝ)) |
| (1 - (J.card : ℝ) ^ (-19 / 20 : ℝ))}) μ) |
| (hmiss : ∀ k ∈ largePhaseTimes N ((J.card : ℝ) ^ ((4 / 5 : ℝ) * J.card)), |
| ∀ i ∈ J, μ {θ : ι → ℝ | 1 - Int.fract ((k : ℝ) * θ i) ∉ |
| Set.Icc ((J.card : ℝ) ^ (-19 / 20 : ℝ)) |
| (1 - (J.card : ℝ) ^ (-19 / 20 : ℝ))} ≤ |
| ENNReal.ofReal (10 * (J.card : ℝ) ^ (-19 / 20 : ℝ))) : |
| μ (largePhaseFailure J N) ≤ ENNReal.ofReal (failureBound J.card) := by |
| classical |
| let m : ℝ := J.card |
| let b : ℝ := m ^ (-19 / 20 : ℝ) |
| let R : ℝ := m ^ ((4 / 5 : ℝ) * m) |
| let q : ℝ := (19 / 20 : ℝ) * m |
| let times := largePhaseTimes N R |
| let E : ℤ → ι → Set (ι → ℝ) := fun k i => |
| {θ | 1 - Int.fract ((k : ℝ) * θ i) ∉ Set.Icc b (1 - b)} |
| have hm0 : 0 ≤ m := by dsimp [m]; positivity |
| have hb : 0 ≤ b := Real.rpow_nonneg hm0 _ |
| have hq : 0 ≤ q := by dsimp [q]; positivity |
| have hp : ENNReal.ofReal (10 * b) ≤ 1 := by |
| rw [← ENNReal.ofReal_one] |
| exact ENNReal.ofReal_le_ofReal (ten_mul_phaseWidth_le_one hm) |
| have hsub : largePhaseFailure J N ⊆ |
| {θ | ∃ k ∈ times, ∃ t : Finset ι, |
| t ⊆ J ∧ q ≤ (t.card : ℝ) ∧ ∀ i ∈ t, θ ∈ E k i} := by |
| intro θ hθ |
| obtain ⟨k, hk, hsmall⟩ := hθ |
| obtain ⟨t, ht, hcard, htmiss⟩ := |
| Invariant233.exists_many_misses_of_few_successes J |
| (fun i => 1 - Int.fract ((k : ℝ) * θ i) ∈ Set.Icc b (1 - b)) hsmall |
| exact ⟨k, hk, t, ht, hcard.le, htmiss⟩ |
| have hprob := Invariant233.measure_exists_many_independent_events_le μ E J times |
| (fun k _ => hind k) hp hmiss q |
| have hnum : (times.card : ℝ) * |
| ((2 : ℝ) ^ J.card * (10 * b) ^ q) ≤ failureBound m := by |
| calc |
| (times.card : ℝ) * ((2 : ℝ) ^ J.card * (10 * b) ^ q) ≤ |
| (2 * R) * ((2 : ℝ) ^ J.card * (10 * b) ^ q) := |
| mul_le_mul_of_nonneg_right (card_largePhaseTimes_le (Real.rpow_nonneg hm0 _)) |
| (by positivity) |
| _ = failureBound m := by |
| dsimp [failureBound, R, b, q, m] |
| rw [Real.rpow_natCast] |
| ring |
| calc |
| μ (largePhaseFailure J N) ≤ μ {θ | ∃ k ∈ times, ∃ t : Finset ι, |
| t ⊆ J ∧ q ≤ (t.card : ℝ) ∧ ∀ i ∈ t, θ ∈ E k i} := measure_mono hsub |
| _ ≤ (times.card : ℝ≥0∞) * ((2 : ℝ≥0∞) ^ J.card * |
| ENNReal.ofReal (10 * b) ^ q) := hprob |
| _ = ENNReal.ofReal ((times.card : ℝ) * ((2 : ℝ) ^ J.card * (10 * b) ^ q)) := by |
| rw [ENNReal.ofReal_mul (p := (times.card : ℝ)) (Nat.cast_nonneg _), |
| ENNReal.ofReal_mul (p := (2 : ℝ) ^ J.card) (by positivity), |
| ENNReal.ofReal_pow (by norm_num), |
| ENNReal.ofReal_rpow_of_nonneg (by positivity) hq] |
| simp |
| _ ≤ ENNReal.ofReal (failureBound m) := ENNReal.ofReal_le_ofReal hnum |
|
|
| theorem iIndepSet_phaseMiss_infinitePi {ι : Type*} (ν : ι → Measure ℝ) |
| [∀ i, IsProbabilityMeasure (ν i)] (b : ℝ) (k : ℤ) : |
| iIndepSet (fun i => {θ : ι → ℝ | 1 - Int.fract ((k : ℝ) * θ i) ∉ |
| Set.Icc b (1 - b)}) (Measure.infinitePi ν) := by |
| have hA : MeasurableSet {x : ℝ | 1 - Int.fract ((k : ℝ) * x) ∉ |
| Set.Icc b (1 - b)} := by |
| exact ((measurable_const.sub (measurable_const.mul measurable_id).fract) |
| measurableSet_Icc).compl |
| have hm : ∀ i : ι, MeasurableSet {θ : ι → ℝ | |
| 1 - Int.fract ((k : ℝ) * θ i) ∉ Set.Icc b (1 - b)} := |
| fun i => hA.preimage (measurable_pi_apply i) |
| apply (iIndepSet_iff_meas_biInter hm).mpr |
| intro S |
| have hi : iIndepFun (fun i (θ : ι → ℝ) => θ i) (Measure.infinitePi ν) := |
| iIndepFun_infinitePi (X := fun _ (x : ℝ) => x) (fun _ => measurable_id) |
| exact hi.measure_inter_preimage_eq_mul S (fun _ _ => hA) |
|
|
| theorem infinitePi_phaseMiss_eq {ι : Type*} (ν : ι → Measure ℝ) |
| [∀ i, IsProbabilityMeasure (ν i)] (b : ℝ) (k : ℤ) (i : ι) : |
| (Measure.infinitePi ν) {θ : ι → ℝ | |
| 1 - Int.fract ((k : ℝ) * θ i) ∉ Set.Icc b (1 - b)} = |
| ν i {x : ℝ | 1 - Int.fract ((k : ℝ) * x) ∉ Set.Icc b (1 - b)} := by |
| have hA : MeasurableSet {x : ℝ | 1 - Int.fract ((k : ℝ) * x) ∉ |
| Set.Icc b (1 - b)} := by |
| exact ((measurable_const.sub (measurable_const.mul measurable_id).fract) |
| measurableSet_Icc).compl |
| simpa only [Measure.infinitePi_map_eval, Set.preimage_ofPred_eq] using |
| (Measure.map_apply (μ := Measure.infinitePi ν) |
| (f := fun θ : ι → ℝ => θ i) (measurable_pi_apply i) hA).symm |
|
|
| theorem ae_largePhaseCondition_of_miss_bounds {ι : Type*} |
| (μ : Measure (ι → ℝ)) [IsFiniteMeasure μ] |
| (J : ℕ → Finset ι) (N : ℕ → ℝ) |
| (hN : ∀ j, 0 < N j) (hgrowth : ∀ j, 2 ^ j ≤ (J j).card) |
| (hind : ∀ (j : ℕ) (k : ℤ), iIndepSet |
| (fun i => {θ : ι → ℝ | 1 - Int.fract ((k : ℝ) * θ i) ∉ |
| Set.Icc (((J j).card : ℝ) ^ (-19 / 20 : ℝ)) |
| (1 - ((J j).card : ℝ) ^ (-19 / 20 : ℝ))}) μ) |
| (hmiss : ∀ᶠ j in atTop, |
| ∀ k ∈ largePhaseTimes (N j) (((J j).card : ℝ) ^ ((4 / 5 : ℝ) * (J j).card)), |
| ∀ i ∈ J j, μ {θ : ι → ℝ | 1 - Int.fract ((k : ℝ) * θ i) ∉ |
| Set.Icc (((J j).card : ℝ) ^ (-19 / 20 : ℝ)) |
| (1 - ((J j).card : ℝ) ^ (-19 / 20 : ℝ))} ≤ |
| ENNReal.ofReal (10 * ((J j).card : ℝ) ^ (-19 / 20 : ℝ))) : |
| ∀ᵐ θ ∂μ, ∀ᶠ j in atTop, largePhaseCondition (J j) (N j) θ := by |
| have hreal : ∀ j : ℕ, (2 : ℝ) ^ j ≤ ((J j).card : ℝ) := by |
| intro j |
| exact_mod_cast hgrowth j |
| have hindex : ∀ j : ℕ, (j : ℝ) ≤ ((J j).card : ℝ) := by |
| intro j |
| exact_mod_cast (le_trans (Nat.lt_two_pow_self (n := j)).le (hgrowth j)) |
| have htop : Tendsto (fun j => ((J j).card : ℝ)) atTop atTop := |
| tendsto_atTop_mono hindex tendsto_natCast_atTop_atTop |
| have hlarge : ∀ᶠ j in atTop, (100 : ℝ) ≤ (J j).card := |
| htop.eventually_ge_atTop 100 |
| have hbad : ∀ᶠ j in atTop, μ (largePhaseFailure (J j) (N j)) ≤ |
| ENNReal.ofReal (failureBound (J j).card) := by |
| filter_upwards [hlarge, hmiss] with j hj hm |
| exact measure_largePhaseFailure_le μ (J j) (N j) hj (hind j) hm |
| have hbc := ae_eventually_notMem_of_phase_bound μ |
| (fun j => largePhaseFailure (J j) (N j)) hreal hbad |
| filter_upwards [hbc] with θ hθ |
| filter_upwards [hθ] with j hj |
| exact (not_mem_largePhaseFailure_iff (J j) (hN j) θ).mp hj |
|
|
| theorem uniformProduct_ae_largePhaseCondition {ι : Type*} |
| (S : ι → ℝ) (hS : ∀ i, 0 < S i) |
| (J : ℕ → Finset ι) (N : ℕ → ℝ) (hN : ∀ j, 0 < N j) |
| (hgrowth : ∀ j, 2 ^ j ≤ (J j).card) |
| (hblock : ∀ j, ∀ i ∈ J j, S i = N j) : |
| ∀ᵐ θ ∂Measure.infinitePi |
| (fun i => ProbabilityTheory.cond volume (Set.Icc (1 / S i) (2 / S i))), |
| ∀ᶠ j in atTop, largePhaseCondition (J j) (N j) θ := by |
| let ν : ι → Measure ℝ := fun i => |
| ProbabilityTheory.cond volume (Set.Icc (1 / S i) (2 / S i)) |
| let (i : ι) : IsProbabilityMeasure (ν i) := |
| InvariantProjection.IndependentAngles.intervalLaw_isProbabilityMeasure |
| (div_lt_div_of_pos_right (by norm_num) (hS i)) |
| apply ae_largePhaseCondition_of_miss_bounds (Measure.infinitePi ν) J N hN hgrowth |
| · intro j k |
| exact iIndepSet_phaseMiss_infinitePi ν _ k |
| · apply Filter.Eventually.of_forall |
| intro j k hk i hi |
| rw [infinitePi_phaseMiss_eq ν _ k i] |
| apply (ENNReal.le_ofReal_iff_toReal_le (measure_ne_top (ν i) _) (by positivity)).mpr |
| change (ProbabilityTheory.cond volume (Set.Icc (1 / S i) (2 / S i))).real |
| {x : ℝ | 1 - Int.fract ((k : ℝ) * x) ∉ |
| Set.Icc (((J j).card : ℝ) ^ (-19 / 20 : ℝ)) |
| (1 - ((J j).card : ℝ) ^ (-19 / 20 : ℝ))} ≤ _ |
| rw [hblock j i hi] |
| exact Invariant233.ModuloIntervalMeasure.uniform_angle_threshold_miss_le |
| (hN j) ((mem_largePhaseTimes (hN j)).mp hk).1 (by positivity) |
|
|
| theorem exists_summable_independent_largePhases {ι : Type*} [Countable ι] |
| (S : ι → ℝ) (hS : ∀ i, 0 < S i) (hsum : Summable (fun i => 1 / S i)) |
| (J : ℕ → Finset ι) (N : ℕ → ℝ) (hN : ∀ j, 0 < N j) |
| (hgrowth : ∀ j, 2 ^ j ≤ (J j).card) |
| (hblock : ∀ j, ∀ i ∈ J j, S i = N j) : |
| ∃ θ : ι → ℝ, |
| (∀ i, θ i ∈ Set.Icc (1 / S i) (2 / S i)) ∧ Summable θ ∧ |
| (∑' i, θ i) ≤ 2 * (∑' i, 1 / S i) ∧ |
| LinearIndependent ℚ (fun o : Option ι => o.elim 1 θ) ∧ |
| (∀ᶠ j in atTop, largePhaseCondition (J j) (N j) θ) := by |
| exact InvariantProjection.IndependentAngles.exists_prescribed_summable_independent_and |
| S hS hsum (uniformProduct_ae_largePhaseCondition S hS J N hN hgrowth hblock) |
|
|
| theorem smallPhaseCondition_of_bounds {ι : Type*} (J : Finset ι) |
| {K G : ℝ} (hJ : 0 < J.card) (hK : 0 < K) (hG : 0 < G) |
| (θ : ι → ℝ) |
| (hθ : ∀ i ∈ J, θ i ∈ Set.Icc (1 / ((J.card : ℝ) * K)) |
| (2 / ((J.card : ℝ) * K))) : |
| smallPhaseCondition J K G θ := by |
| intro k hklo hkhi i hi |
| exact Invariant233.phase_small_window |
| (by exact_mod_cast hJ) hK hG (hθ i hi).1 (hθ i hi).2 hklo hkhi |
|
|
| theorem uniformProduct_ae_phaseRanges {ι : Type*} [Countable ι] |
| (S : ι → ℝ) (hS : ∀ i, 0 < S i) |
| (J : ℕ → Finset ι) (K G : ℕ → ℝ) |
| (hK : ∀ j, 0 < K j) (hG : ∀ j, 0 < G j) |
| (hgrowth : ∀ j, 2 ^ j ≤ (J j).card) |
| (hblock : ∀ j, ∀ i ∈ J j, S i = ((J j).card : ℝ) * K j) : |
| ∀ᵐ θ ∂Measure.infinitePi |
| (fun i => ProbabilityTheory.cond volume (Set.Icc (1 / S i) (2 / S i))), |
| (∀ j, smallPhaseCondition (J j) (K j) (G j) θ) ∧ |
| (∀ᶠ j in atTop, largePhaseCondition (J j) (((J j).card : ℝ) * K j) θ) := by |
| have hJ : ∀ j, 0 < (J j).card := fun j => |
| lt_of_lt_of_le (by positivity : 0 < 2 ^ j) (hgrowth j) |
| have hN : ∀ j, 0 < ((J j).card : ℝ) * K j := fun j => |
| mul_pos (by exact_mod_cast hJ j) (hK j) |
| have hlarge := uniformProduct_ae_largePhaseCondition S hS J |
| (fun j => ((J j).card : ℝ) * K j) hN hgrowth hblock |
| have hbounds := InvariantProjection.IndependentAngles.intervalProduct_ae_bounds_linearIndependent |
| (fun i => 1 / S i) (fun i => 2 / S i) |
| (fun i => div_lt_div_of_pos_right (by norm_num) (hS i)) |
| filter_upwards [hlarge, hbounds] with θ hθlarge hθbounds |
| refine ⟨?_, hθlarge⟩ |
| intro j |
| apply smallPhaseCondition_of_bounds (J j) (hJ j) (hK j) (hG j) θ |
| intro i hi |
| simpa only [hblock j i hi] using hθbounds.1 i |
|
|
| theorem exists_summable_independent_phaseRanges {ι : Type*} [Countable ι] |
| (S : ι → ℝ) (hS : ∀ i, 0 < S i) (hsum : Summable (fun i => 1 / S i)) |
| (J : ℕ → Finset ι) (K G : ℕ → ℝ) |
| (hK : ∀ j, 0 < K j) (hG : ∀ j, 0 < G j) |
| (hgrowth : ∀ j, 2 ^ j ≤ (J j).card) |
| (hblock : ∀ j, ∀ i ∈ J j, S i = ((J j).card : ℝ) * K j) : |
| ∃ θ : ι → ℝ, |
| (∀ i, θ i ∈ Set.Icc (1 / S i) (2 / S i)) ∧ Summable θ ∧ |
| (∑' i, θ i) ≤ 2 * (∑' i, 1 / S i) ∧ |
| LinearIndependent ℚ (fun o : Option ι => o.elim 1 θ) ∧ |
| (∀ j, smallPhaseCondition (J j) (K j) (G j) θ) ∧ |
| (∀ᶠ j in atTop, largePhaseCondition (J j) (((J j).card : ℝ) * K j) θ) := by |
| exact InvariantProjection.IndependentAngles.exists_prescribed_summable_independent_and |
| S hS hsum (uniformProduct_ae_phaseRanges S hS J K G hK hG hgrowth hblock) |
|
|
| end Invariant233.ProductPhases |
|
|
| end OAI |
|
|
