openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Sampling /CircleDigitEndpointLoss.lean
| import OAI.Analysis.IrrationalRotation.Rotation.CircleQuantizedMarginals | |
| import OAI.Analysis.IrrationalRotation.Analysis.EndpointLoss | |
| import OAI.Analysis.IrrationalRotation.Sampling.CircleDigitMarginalLaws | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory Filter | |
| open scoped ENNReal Topology | |
| namespace Invariant233.CircleCoupling | |
| variable {θ : ℝ} {p : CircleParameterData} | |
| {hr : ∀ i, 0 < MixedRadix.radices p.q i} | |
| namespace DigitMarginalLaws | |
| lemma sqrt_integral_sq_le (hM : DigitMarginalLaws p hr) | |
| {H : Circle → ℝ} (hH : MemLp H 2 volume) | |
| (hbound : eLpNorm H 2 volume ≤ ENNReal.ofReal marginalBound) : | |
| Real.sqrt (∫ x, H x ^ 2) ≤ marginalBound := by | |
| rw [EndpointLoss.sqrt_integral_sq_eq_toReal_eLpNorm hH] | |
| have ht := ENNReal.toReal_mono ENNReal.ofReal_ne_top hbound | |
| simpa only [ENNReal.toReal_ofReal hM.1.le] using ht | |
| lemma hybrid_preimage_measureReal_le (hM : DigitMarginalLaws p hr) | |
| (hp : CircleParameterLaws θ p) | |
| (j : ℕ) (ε : ℕ → Bool) {s : Set Circle} (hs : MeasurableSet s) : | |
| (law p j).real (hybridPoint p hr ε ⁻¹' s) ≤ | |
| marginalBound * Real.sqrt (volume.real s) := by | |
| obtain ⟨H, hH, hnonneg, _hmean, hbound, hlaw⟩ := hM.2 j ε | |
| have hmass := EndpointLoss.withDensity_measureReal_le_sqrt | |
| (volume : Measure Circle) hH hnonneg hs | |
| rw [← hlaw, map_measureReal_apply (measurable_hybridPoint hp hr ε) hs] at hmass | |
| exact hmass.trans (mul_le_mul_of_nonneg_right | |
| (hM.sqrt_integral_sq_le hH hbound) (Real.sqrt_nonneg _)) | |
| theorem tendsto_hybrid_sublevel_zero (hM : DigitMarginalLaws p hr) | |
| (hp : CircleParameterLaws θ p) | |
| (ε : ℕ → ℕ → Bool) {r : Circle → ℝ} | |
| (hrmeas : Measurable r) (hrpos : ∀ᵐ x ∂volume, 0 < r x) : | |
| Tendsto (fun j : ℕ => (law p j).real | |
| {ω | r (hybridPoint p hr (ε j) ω) < 1 / (j : ℝ)}) | |
| atTop (𝓝 0) := by | |
| have hbase := EndpointLoss.tendsto_measureReal_one_div_sublevel_zero | |
| (volume : Measure Circle) hrmeas hrpos | |
| apply squeeze_zero (fun _ => measureReal_nonneg) | |
| (fun j => hM.hybrid_preimage_measureReal_le hp j (ε j) | |
| (measurableSet_lt hrmeas measurable_const)) | |
| simpa only [Function.comp_apply, Real.sqrt_zero, mul_zero] using | |
| tendsto_const_nhds.mul (Real.continuous_sqrt.continuousAt.tendsto.comp hbase) | |
| theorem tendsto_endpoint_bad_union_zero (hM : DigitMarginalLaws p hr) | |
| (hp : CircleParameterLaws θ p) | |
| {r : Circle → ℝ} (hrmeas : Measurable r) | |
| (hrinterior : ∀ᵐ x ∂volume, 0 < r x ∧ r x < 1) : | |
| Tendsto (fun j : ℕ => (law p j).real | |
| ({ω | r (point p hr false ω) < 1 / (j : ℝ)} ∪ | |
| {ω | 1 - r (point p hr true ω) < 1 / (j : ℝ)})) | |
| atTop (𝓝 0) := by | |
| have hleft := hM.tendsto_hybrid_sublevel_zero hp (fun _ _ => false) | |
| hrmeas (hrinterior.mono fun _ hx => hx.1) | |
| have hright := hM.tendsto_hybrid_sublevel_zero hp (fun _ _ => true) | |
| (measurable_const.sub hrmeas) | |
| (hrinterior.mono fun _ hx => sub_pos.mpr hx.2) | |
| apply squeeze_zero (fun _ => measureReal_nonneg) | |
| (fun _ => measureReal_union_le _ _) | |
| simpa only [hybridPoint_const, Pi.sub_apply, zero_add] using hleft.add hright | |
| end DigitMarginalLaws | |
| theorem tendsto_endpoint_bad_union_zero_of_parameterLaws | |
| (hp : CircleParameterLaws θ p) | |
| (hr : ∀ i, 0 < MixedRadix.radices p.q i) | |
| {r : Circle → ℝ} (hrmeas : Measurable r) | |
| (hrinterior : ∀ᵐ x ∂volume, 0 < r x ∧ r x < 1) : | |
| Tendsto (fun j : ℕ => (law p j).real | |
| ({ω | r (point p hr false ω) < 1 / (j : ℝ)} ∪ | |
| {ω | 1 - r (point p hr true ω) < 1 / (j : ℝ)})) | |
| atTop (𝓝 0) := | |
| (digitMarginalLaws hp hr).tendsto_endpoint_bad_union_zero hp hrmeas hrinterior | |
| end Invariant233.CircleCoupling | |
| end | |
| end OAI | |