File size: 3,408 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.CharacterSamplingTests
import OAI.Analysis.IrrationalRotation.Sampling.CharacterPhaseRealization
namespace OAI
noncomputable section
open MeasureTheory Filter Set
open scoped Topology BigOperators
namespace Invariant233.CharacterSampling
theorem exists_realized_sampling_block {θ : ℝ} (hθ : Irrational θ)
{m : ℕ} (hm : 200 ≤ m) (target : Fin m → ℝ)
(htarget0 : ∀ i, 0 < target i) (htarget1 : ∀ i, target i < 1)
{q₀ : ℕ} (hq₀ : 0 < q₀) (R : ℝ)
(U : Set (Fin m → ℝ)) (hU : U ∈ 𝓝 target)
(r : ℕ) (powers : Fin r → Finset ℕ) (δ : Fin r → Bool → ℝ)
(hcut : ∀ k, ∀ n ∈ powers k, ∀ i, Int.fract ((n : ℝ) * target i) ≠ 0)
(hbound : ∀ k endpoint,
(∫ y : UnitAddTorus (Fin m), squaredError m (powers k) endpoint (target,y)) < δ k endpoint) :
∃ q : ℕ → ℕ,
q 0 = q₀ ∧ (∀ i, 0 < q i) ∧ StrictMono q ∧
(∀ i, q i ∣ q (i + 1)) ∧
(∀ i < m, R ≤ (q (i + 1) : ℝ) / (q i : ℝ)) ∧
(fun i : Fin m => Int.fract ((q (i.val + 1) : ℝ) * θ)) ∈ U ∧
∀ k endpoint,
(∫ x : UnitAddCircle, squaredError m (powers k) endpoint
((fun i : Fin m => Int.fract ((q (i.val + 1) : ℝ) * θ)),
fun i : Fin m => (q (i.val + 1) : ℤ) • x)) < δ k endpoint := by
classical
let target' : ℕ → ℝ := fun i => if h : i < m then target ⟨i,h⟩ else 1/2
have ht0 : ∀ i, 0 < target' i := by
intro i
dsimp only [target']
split_ifs with hi
· exact htarget0 ⟨i,hi⟩
· norm_num
have ht1 : ∀ i, target' i < 1 := by
intro i
dsimp only [target']
split_ifs with hi
· exact htarget1 ⟨i,hi⟩
· norm_num
obtain ⟨Q, hzero, hpos, hstrict, hdiv, hgap, hphase'⟩ :=
PrescribedPhaseRealization.exists_phase_approximation_family hθ target' ht0 ht1 hq₀
let phase : ℕ → Fin m → ℝ := fun j i => Int.fract ((Q j (i.val + 1) : ℝ) * θ)
have hphase : Tendsto phase atTop (𝓝 target) := by
apply tendsto_pi_nhds.mpr
intro i
simpa only [phase, target', dite_eq_left i.isLt] using hphase' i.val
have hratios : ∀ᶠ j in atTop, ∀ i < m,
R ≤ (Q j (i + 1) : ℝ) / (Q j i : ℝ) := by
have hfinite : ∀ᶠ j in atTop, ∀ i ∈ Finset.range m,
R ≤ (Q j (i + 1) : ℝ) / (Q j i : ℝ) := by
rw [Finset.eventually_all]
intro i hi
exact (hgap i).eventually_ge_atTop R
simpa only [Finset.mem_range] using hfinite
have htests : ∀ᶠ j in atTop, ∀ k endpoint,
(∫ x : UnitAddCircle, squaredError m (powers k) endpoint
(phase j, fun i : Fin m => (Q j (i.val + 1) : ℤ) • x)) < δ k endpoint := by
apply Filter.eventually_all.mpr
intro k
apply Filter.eventually_all.mpr
intro endpoint
have hlim := tendsto_integral_squaredError hm (powers k) endpoint
(fun j i => Q j (i + 1)) (fun j i => hpos j (i + 1))
(fun j i l hil => (hstrict j).monotone (Nat.add_le_add_right hil 1))
(fun i _ => hgap (i + 1)) hphase (hcut k)
exact hlim.eventually_lt_const (hbound k endpoint)
obtain ⟨j, ⟨⟨hjratio, hjU⟩, hjtests⟩⟩ :=
((hratios.and (hphase.eventually hU)).and htests).exists
exact ⟨Q j, hzero j, hpos j, hstrict j, hdiv j, hjratio, hjU, hjtests⟩
end Invariant233.CharacterSampling
end
end OAI
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