openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Sampling /CircleSamplingGraphLimits.lean
| import OAI.Analysis.IrrationalRotation.Sampling.CircleSamplingGraph | |
| import OAI.Analysis.IrrationalRotation.Analysis.GraphMeasureConvergence | |
| import OAI.Analysis.IrrationalRotation.Analysis.L2UniformAbsoluteContinuity | |
| import OAI.Analysis.IrrationalRotation.Sampling.CirclePhaseTails | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory Filter Set | |
| open scoped Topology ENNReal | |
| namespace Invariant233.CircleParameterData | |
| def circleFiniteBase : FiniteMeasure Circle := | |
| ⟨circleMeasure, by unfold circleMeasure; infer_instance⟩ | |
| instance circleFiniteBase_probability : IsProbabilityMeasure (circleFiniteBase : Measure Circle) := by | |
| constructor | |
| change volume (Set.univ : Set Circle) = 1 | |
| exact UnitAddCircle.measure_univ | |
| def samplingGraphFinite (p : CircleParameterData) (θ : ℝ) (j r : ℕ) | |
| (hm : 0 < p.m j) : FiniteMeasure (Circle × Circle) := | |
| ⟨p.samplingGraphAverage θ j r, p.samplingGraphAverage_finite θ j r hm⟩ | |
| end Invariant233.CircleParameterData | |
| namespace Invariant233.CircleParameterLaws | |
| open CircleParameterData | |
| variable {θ : ℝ} {p : CircleParameterData} (hp : CircleParameterLaws θ p) | |
| include hp | |
| theorem samplingAverage_error_tendsto_zero | |
| (j r : ℕ → ℕ) (hj : ∀ n, 1 ≤ j n) | |
| (hr : ∀ n, r n < p.q (p.blockEnd (j n - 1))) | |
| (hjt : Tendsto j atTop atTop) (endpoint : Bool) : | |
| Tendsto (fun n => eLpNorm (fun x => p.samplingAverage θ (j n) (r n) endpoint x - 1) | |
| 2 circleMeasure) atTop (𝓝 0) := by | |
| have hlim := ENNReal.continuous_ofReal.continuousAt.tendsto.comp | |
| (tendsto_one_div_atTop_nhds_zero_nat.comp hjt) | |
| apply tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds | |
| (show Tendsto (fun n => ENNReal.ofReal (1 / (j n : ℝ))) atTop (𝓝 0) from by | |
| simpa only [Function.comp_def, ENNReal.ofReal_zero] using hlim) | |
| (fun _ => bot_le) | |
| (fun n => (hp.sampling_L2 (j n) (hj n) (r n) (hr n) endpoint).le) | |
| theorem tendsto_samplingGraph_first_marginal | |
| (j r : ℕ → ℕ) (hj : ∀ n, 1 ≤ j n) | |
| (hr : ∀ n, r n < p.q (p.blockEnd (j n - 1))) | |
| (hjt : Tendsto j atTop atTop) : | |
| Tendsto (fun n => (p.samplingGraphFinite θ (j n) (r n) (hp.m_pos _ (hj n))).map Prod.fst) | |
| atTop (𝓝 circleFiniteBase) := by | |
| apply GraphMeasureConvergence.tendsto_of_density_L2 circleFiniteBase _ | |
| (fun n => p.samplingAverage θ (j n) (r n) false) | |
| · intro n | |
| exact p.memLp_samplingAverage θ (j n) (r n) false (hp.m_pos _ (hj n)) 2 | |
| · intro n | |
| exact Eventually.of_forall fun x => | |
| (p.samplingAverage_bounds θ (j n) (r n) false (hp.m_pos _ (hj n)) x).1 | |
| · intro n | |
| exact p.samplingGraphAverage_first_marginal θ (j n) (r n) (hp.m_pos _ (hj n)) | |
| · exact hp.samplingAverage_error_tendsto_zero j r hj hr hjt false | |
| theorem tendsto_samplingGraph_second_marginal | |
| (j r : ℕ → ℕ) (hj : ∀ n, 1 ≤ j n) | |
| (hr : ∀ n, r n < p.q (p.blockEnd (j n - 1))) | |
| (hjt : Tendsto j atTop atTop) : | |
| Tendsto (fun n => (p.samplingGraphFinite θ (j n) (r n) (hp.m_pos _ (hj n))).map Prod.snd) | |
| atTop (𝓝 circleFiniteBase) := by | |
| apply GraphMeasureConvergence.tendsto_of_density_L2 circleFiniteBase _ | |
| (fun n => p.samplingAverage θ (j n) (r n) true) | |
| · intro n | |
| exact p.memLp_samplingAverage θ (j n) (r n) true (hp.m_pos _ (hj n)) 2 | |
| · intro n | |
| exact Eventually.of_forall fun x => | |
| (p.samplingAverage_bounds θ (j n) (r n) true (hp.m_pos _ (hj n)) x).1 | |
| · intro n | |
| exact p.samplingGraphAverage_second_marginal θ (j n) (r n) (hp.m_pos _ (hj n)) | |
| · exact hp.samplingAverage_error_tendsto_zero j r hj hr hjt true | |
| theorem samplingGraph_marginals_uniformlyAbsolutelyContinuous | |
| (j r : ℕ → ℕ) (hj : ∀ n, 1 ≤ j n) | |
| (hr : ∀ n, r n < p.q (p.blockEnd (j n - 1))) : | |
| UniformlyAbsolutelyContinuous circleMeasure | |
| (fun n => (p.samplingGraphAverage θ (j n) (r n)).map Prod.fst) ∧ | |
| UniformlyAbsolutelyContinuous circleMeasure | |
| (fun n => (p.samplingGraphAverage θ (j n) (r n)).map Prod.snd) := by | |
| let : IsFiniteMeasure circleMeasure := by unfold circleMeasure; infer_instance | |
| have hden (endpoint : Bool) : UniformlyAbsolutelyContinuous circleMeasure | |
| (fun n => circleMeasure.withDensity | |
| (fun x => ENNReal.ofReal (p.samplingAverage θ (j n) (r n) endpoint x))) := by | |
| apply EndpointLoss.uniformlyAbsolutelyContinuous_withDensity circleMeasure | |
| (fun n => p.samplingAverage θ (j n) (r n) endpoint) | |
| (fun n => p.memLp_samplingAverage θ (j n) (r n) endpoint (hp.m_pos _ (hj n)) 2) | |
| (fun n => Eventually.of_forall fun x => | |
| (p.samplingAverage_bounds θ (j n) (r n) endpoint (hp.m_pos _ (hj n)) x).1) 2 | |
| intro n | |
| rw [EndpointLoss.sqrt_integral_sq_eq_toReal_eLpNorm | |
| (p.memLp_samplingAverage θ (j n) (r n) endpoint (hp.m_pos _ (hj n)) 2)] | |
| have h := ENNReal.toReal_mono (by norm_num : (2 : ENNReal) ≠ ∞) | |
| (hp.samplingAverage_eLpNorm_le_two (j n) (hj n) (r n) (hr n) endpoint) | |
| simpa only [ENNReal.toReal_ofNat] using h | |
| constructor | |
| · convert hden false using 1 | |
| funext n | |
| exact p.samplingGraphAverage_first_marginal θ (j n) (r n) (hp.m_pos _ (hj n)) | |
| · convert hden true using 1 | |
| funext n | |
| exact p.samplingGraphAverage_second_marginal θ (j n) (r n) (hp.m_pos _ (hj n)) | |
| theorem ell_nonneg_and_le (j : ℕ) : 0 ≤ p.ell j ∧ | |
| p.ell j ≤ 1 / ((j + 1 : ℕ) : ℝ)^2 := by | |
| have hq : (1 : ℝ) ≤ p.q (p.blockEnd (j - 1)) := by | |
| exact_mod_cast hp.q_pos (p.blockEnd (j - 1)) | |
| have ha : 0 ≤ (p.blockEnd (j - 1) : ℝ) := Nat.cast_nonneg _ | |
| have hj : (0 : ℝ) < ((j + 1 : ℕ) : ℝ)^2 := by positivity | |
| have hprod : 1 ≤ (1 + (p.blockEnd (j - 1) : ℝ)) * | |
| (p.q (p.blockEnd (j - 1)) : ℝ) := | |
| one_le_mul_of_one_le_of_one_le (by linarith) hq | |
| unfold ell | |
| constructor | |
| · positivity | |
| · apply one_div_le_one_div_of_le hj | |
| nlinarith | |
| theorem tendsto_ell_zero : Tendsto p.ell atTop (𝓝 0) := by | |
| have h : Tendsto (fun n : ℕ => 1 / ((n + 1 : ℕ) : ℝ)) atTop (𝓝 (0 : ℝ)) := | |
| (tendsto_add_atTop_iff_nat 1).mpr tendsto_one_div_atTop_nhds_zero_nat | |
| have hsq : Tendsto (fun n : ℕ => 1 / ((n + 1 : ℕ) : ℝ)^2) atTop (𝓝 (0 : ℝ)) := by | |
| simpa only [one_div_pow, zero_pow (by decide : 2 ≠ 0)] using h.pow 2 | |
| exact squeeze_zero (fun n => (hp.ell_nonneg_and_le n).1) | |
| (fun n => (hp.ell_nonneg_and_le n).2) hsq | |
| theorem samplingGraphAverage_ae_dist_lt | |
| (j : ℕ) (hj : 1 ≤ j) (r : ℕ) | |
| (hr : r < p.q (p.blockEnd (j - 1))) (t : Circle) | |
| (hshift : (p.shift j r : Circle) = t) : | |
| ∀ᵐ z ∂p.samplingGraphAverage θ j r, dist z.2 (z.1 + t) < p.ell j := by | |
| filter_upwards [p.samplingGraphAverage_ae_graph θ j r] with z hz | |
| obtain ⟨k, hk, heq⟩ := hz | |
| have hphase := (hp.samples_phase j hj r hr k hk).2 | |
| rw [heq] | |
| change dist (z.1 + (((k : ℝ) * θ : ℝ) : Circle)) (z.1 + t) < p.ell j | |
| rw [dist_add_left, ← hshift, dist_eq_norm, ← AddCircle.coe_sub] | |
| calc | |
| ‖(((k : ℝ) * θ - p.shift j r : ℝ) : Circle)‖ = | |
| ‖((Int.fract ((k : ℝ) * θ - p.shift j r) : ℝ) : Circle)‖ := by | |
| rw [AddCircle.coe_fract] | |
| _ ≤ ‖Int.fract ((k : ℝ) * θ - p.shift j r)‖ := QuotientAddGroup.norm_mk_le_norm | |
| _ = Int.fract ((k : ℝ) * θ - p.shift j r) := | |
| Real.norm_of_nonneg (Int.fract_nonneg _) | |
| _ < p.ell j := hphase | |
| theorem tendsto_samplingGraph_translation | |
| (j r : ℕ → ℕ) (hj : ∀ n, 1 ≤ j n) | |
| (hr : ∀ n, r n < p.q (p.blockEnd (j n - 1))) | |
| (hjt : Tendsto j atTop atTop) (t : Circle) | |
| (hshift : ∀ n, (p.shift (j n) (r n) : Circle) = t) : | |
| Tendsto (fun n => p.samplingGraphFinite θ (j n) (r n) (hp.m_pos _ (hj n))) | |
| atTop (𝓝 (circleFiniteBase.map (fun x => (x, x + t)))) := by | |
| apply GraphMeasureConvergence.tendsto_graph_of_first_marginal _ circleFiniteBase | |
| (fun x => x + t) (continuous_id.add continuous_const) | |
| (hp.tendsto_samplingGraph_first_marginal j r hj hr hjt) | |
| intro δ hδ | |
| filter_upwards [(hp.tendsto_ell_zero.comp hjt).eventually_lt_const hδ] with n hn | |
| exact (hp.samplingGraphAverage_ae_dist_lt (j n) (hj n) (r n) (hr n) t (hshift n)).mono | |
| (fun _ hz => hz.trans hn) | |
| theorem tendsto_samplingGraph_translation_trimmed | |
| (j r : ℕ → ℕ) (hj : ∀ n, 1 ≤ j n) | |
| (hr : ∀ n, r n < p.q (p.blockEnd (j n - 1))) | |
| (hjt : Tendsto j atTop atTop) (t : Circle) | |
| (hshift : ∀ n, (p.shift (j n) (r n) : Circle) = t) | |
| (s : ℕ → Set Circle) (hs : ∀ n, MeasurableSet (s n)) | |
| (hbase : Tendsto (fun n => circleMeasure.real (s n)ᶜ) atTop (𝓝 0)) : | |
| Tendsto (fun n => (p.samplingGraphFinite θ (j n) (r n) (hp.m_pos _ (hj n))).restrict | |
| (Prod.fst ⁻¹' s n)) atTop (𝓝 (circleFiniteBase.map (fun x => (x, x + t)))) := by | |
| apply GraphMeasureConvergence.tendsto_trimmed_graph_of_uniform_first_marginal | |
| _ circleFiniteBase (fun x => x + t) (continuous_id.add continuous_const) | |
| (hp.tendsto_samplingGraph_first_marginal j r hj hr hjt) _ | |
| (hp.samplingGraph_marginals_uniformlyAbsolutelyContinuous j r hj hr).1 s hs hbase | |
| intro δ hδ | |
| filter_upwards [(hp.tendsto_ell_zero.comp hjt).eventually_lt_const hδ] with n hn | |
| exact (hp.samplingGraphAverage_ae_dist_lt (j n) (hj n) (r n) (hr n) t (hshift n)).mono | |
| (fun _ hz => hz.trans hn) | |
| end Invariant233.CircleParameterLaws | |
| end | |
| end OAI | |