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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
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