| import OAI.Analysis.FiniteFactor.OrbitSampling.CylinderSystem |
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| namespace OAI |
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| namespace FiniteFactor |
| open MeasureTheory Filter Finset |
| open scoped Classical ENNReal Topology |
| noncomputable section |
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| theorem rational_Icc_nhds {x : ℝ} {u : Set ℝ} (hu : u ∈ 𝓝 x) : |
| ∃ a b : ℚ, Set.Icc (a : ℝ) b ∈ 𝓝 x ∧ Set.Icc (a : ℝ) b ⊆ u := by |
| obtain ⟨l,r,hlr,hsub⟩ := mem_nhds_iff_exists_Ioo_subset.mp hu |
| obtain ⟨a,hla,hax⟩ := exists_rat_btwn hlr.1 |
| obtain ⟨b,hxb,hbr⟩ := exists_rat_btwn hlr.2 |
| exact ⟨a,b,Icc_mem_nhds hax hxb,fun y hy => hsub ⟨hla.trans_le hy.1,hy.2.trans_lt hbr⟩⟩ |
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| /-- The finite rational cylinders contain arbitrarily small neighborhoods |
| in the full real product, including points outside the unit cube. -/ |
| theorem sourceSamplingCylinder_basis (u : Set SourceSamplingPair) (hu : IsOpen u) |
| (p : SourceSamplingPair) (hp : p ∈ u) : |
| ∃ s ∈ Set.range sourceSamplingCylinder, s ∈ 𝓝 p ∧ s ⊆ u := by |
| obtain ⟨ux,hux,uy,huy,hxy⟩ := mem_nhds_prod_iff.mp (hu.mem_nhds hp) |
| rw [nhds_pi,Filter.mem_pi'] at hux huy |
| obtain ⟨Kx,tx,htx,hsubx⟩ := hux |
| obtain ⟨Ky,ty,hty,hsuby⟩ := huy |
| choose ax bx hnx hsx using fun i => rational_Icc_nhds (htx i) |
| choose ay by' hny hsy using fun i => rational_Icc_nhds (hty i) |
| let T : SourceSamplingRationalTest := ⟨Kx ∪ Ky,fun i => ((ax i,bx i),(ay i,by' i))⟩ |
| have hTx (i : SourceCoordinate) (hi : i ∈ T.1) : |
| sourceSamplingRationalX T i = Set.Icc (ax i : ℝ) (bx i) := by |
| simp [sourceSamplingRationalX,sourceSamplingRationalEndpoints,hi,T] |
| have hTy (i : SourceCoordinate) (hi : i ∈ T.1) : |
| sourceSamplingRationalY T i = Set.Icc (ay i : ℝ) (by' i) := by |
| simp [sourceSamplingRationalY,sourceSamplingRationalEndpoints,hi,T] |
| refine ⟨sourceSamplingCylinder T,⟨T,rfl⟩,?_,?_⟩ |
| · have he : sourceSamplingCylinder T = ⋂ i ∈ T.1, |
| {q : SourceSamplingPair | q.1 i ∈ sourceSamplingRationalX T i ∧ |
| q.2 i ∈ sourceSamplingRationalY T i} := by ext q; simp [sourceSamplingCylinder] |
| rw [he] |
| apply (Filter.biInter_mem (Set.toFinite (T.1 : Set SourceCoordinate))).mpr |
| intro i hi |
| have hx := ((continuous_apply i).comp continuous_fst).continuousAt (hnx i) |
| have hy := ((continuous_apply i).comp continuous_snd).continuousAt (hny i) |
| change {q : SourceSamplingPair | q.1 i ∈ Set.Icc (ax i : ℝ) (bx i)} ∈ 𝓝 p at hx |
| change {q : SourceSamplingPair | q.2 i ∈ Set.Icc (ay i : ℝ) (by' i)} ∈ 𝓝 p at hy |
| filter_upwards [hx,hy] with q hq1 hq2 |
| exact ⟨by simpa only [hTx i hi] using hq1,by simpa only [hTy i hi] using hq2⟩ |
| · intro q hq |
| apply hxy |
| constructor |
| · apply hsubx |
| intro i hi |
| have hmem : i ∈ T.1 := Finset.mem_union_left _ hi |
| exact hsx i (by simpa only [hTx i hmem] using (hq i hmem).1) |
| · apply hsuby |
| intro i hi |
| have hmem : i ∈ T.1 := Finset.mem_union_right _ hi |
| exact hsy i (by simpa only [hTy i hmem] using (hq i hmem).2) |
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| end |
| end FiniteFactor |
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| end OAI |
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