File size: 3,640 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 | import OAI.Analysis.IrrationalRotation.Sampling.CharacterSamplingLp
import OAI.Analysis.IrrationalRotation.Sampling.CharacterSamplingRealization
namespace OAI
noncomputable section
open MeasureTheory Filter Set
open scoped ENNReal Topology
namespace Invariant233.CharacterSampling
local instance characterSamplingLpRealizationCircleProbability :
IsProbabilityMeasure (volume : Measure UnitAddCircle) :=
⟨by simp [AddCircle.measure_univ]⟩
theorem exists_realized_sampling_block_L2 {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)
(hδ : ∀ k endpoint, 0 < δ k endpoint)
(hcut : ∀ k, ∀ n ∈ powers k, ∀ i, Int.fract ((n : ℝ) * target i) ≠ 0)
(hbound : ∀ k endpoint,
eLpNorm (fun y : UnitAddTorus (Fin m) =>
average m (powers k) endpoint target y - 1) 2 volume < ENNReal.ofReal (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,
eLpNorm (fun x : UnitAddCircle => average 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) - 1) 2 volume <
ENNReal.ofReal (δ k endpoint) := by
have hsq : ∀ k endpoint,
(∫ y : UnitAddTorus (Fin m), squaredError m (powers k) endpoint (target, y)) <
(δ k endpoint) ^ 2 := by
intro k endpoint
have hint := integrable_squaredError_comp (μ := volume) hm (powers k) endpoint
(F := fun y : UnitAddTorus (Fin m) => (target, y))
(measurable_const.prodMk measurable_id)
exact (SquaredIntegralLp.eLpNorm_two_lt_ofReal_iff_integral_sq_lt volume
(fun y : UnitAddTorus (Fin m) => average m (powers k) endpoint target y - 1)
(((measurable_average m (powers k) endpoint).comp
(measurable_const.prodMk measurable_id)).sub_const 1).aestronglyMeasurable
hint (hδ k endpoint)).mp (hbound k endpoint)
obtain ⟨q, hzero, hpos, hstrict, hdiv, hratio, hUq, hsqQ⟩ :=
exists_realized_sampling_block hθ hm target htarget0 htarget1 hq₀ R U hU
r powers (fun k endpoint => (δ k endpoint) ^ 2) hcut hsq
refine ⟨q, hzero, hpos, hstrict, hdiv, hratio, hUq, ?_⟩
intro k endpoint
have hc : Continuous (fun x : UnitAddCircle =>
((fun i : Fin m => Int.fract ((q (i.val + 1) : ℝ) * θ)),
fun i : Fin m => (q (i.val + 1) : ℤ) • x)) :=
continuous_const.prodMk (continuous_pi fun i => continuous_zsmul _)
have hint := integrable_squaredError_comp (μ := volume) hm (powers k) endpoint
(F := fun x : UnitAddCircle =>
((fun i : Fin m => Int.fract ((q (i.val + 1) : ℝ) * θ)),
fun i : Fin m => (q (i.val + 1) : ℤ) • x)) hc.measurable
exact (SquaredIntegralLp.eLpNorm_two_lt_ofReal_iff_integral_sq_lt volume
(fun x : UnitAddCircle => average 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) - 1)
(((measurable_average m (powers k) endpoint).comp
hc.measurable).sub_const 1).aestronglyMeasurable
hint (hδ k endpoint)).mpr (hsqQ k endpoint)
end Invariant233.CharacterSampling
end
end OAI
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