| import OAI.Analysis.FiniteFactor.OrbitSampling.RectangleError |
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|
| namespace OAI |
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| namespace FiniteFactor |
| open MeasureTheory Filter Finset |
| open scoped Classical ENNReal Topology |
| noncomputable section |
|
|
| /-- Actual rational rectangle masses approach their ideal product integrals |
| on a single conull event. This is the stochastic approximation step; the |
| identification of the ideal limiting law is a separate obligation. -/ |
| theorem sourceSampling_rectangle_approximation_event : |
| ∀ᵐ θ ∂sourceAngleLaw, ∀ (I : Finset SourceCoordinate) (T : SourceSamplingRationalTest), |
| Tendsto (fun stage => |
| ((sourceSamplingUntrimmed I stage θ) |
| (sourceSamplingRectangle (sourceSamplingRationalX T) (sourceSamplingRationalY T))).toReal - |
| ∫ x, sourceSamplingIdealDensity stage false (fun i => decide (i ∈ I)) |
| (sourceSamplingRationalX T) (sourceSamplingRationalY T) x ∂sourceSamplingHaar) |
| atTop (𝓝 0) := by |
| filter_upwards [sourceSampling_rational_event] with θ hθ |
| intro I T |
| have he := (hθ I false T).1 |
| have hs := Real.continuous_sqrt.continuousAt.tendsto.comp he |
| simp only [Real.sqrt_zero] at hs |
| apply tendsto_zero_iff_norm_tendsto_zero.mpr |
| simp only [Real.norm_eq_abs] |
| apply squeeze_zero' (Eventually.of_forall (fun stage => abs_nonneg _)) _ hs |
| filter_upwards [sourceSamplingPrefix_eventually_covers T.1] with stage hstage |
| exact Real.abs_le_sqrt (sourceSamplingUntrimmed_rectangle_error I stage |
| (sourceSamplingRationalX T) (sourceSamplingRationalY T) |
| (fun _ => measurableSet_Icc) (fun _ => measurableSet_Icc) |
| (sourceSamplingRationalTests_outside T stage hstage) θ) |
|
|
| end |
| end FiniteFactor |
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| end OAI |
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|
