openAI-math-dataset / lean /OAI /NumberTheory /DukePrimeDegree /Geometry /DenseDiagonalGenerators.lean
| import Mathlib.Topology.Instances.AddCircle.DenseSubgroup | |
| import Mathlib.NumberTheory.Real.Irrational | |
| import Mathlib.LinearAlgebra.Pi | |
| import Mathlib.Topology.Algebra.Group.Subgroup | |
| import Mathlib.Topology.Algebra.Constructions.DomMulAct | |
| namespace OAI | |
| /-! | |
| # A finite dense generating family for diagonal parameters | |
| In each real logarithmic coordinate, shifts by one and by square root | |
| of two generate a dense additive subgroup. Thus the diagonal action | |
| has a concrete finite topologically generating family; no one of its | |
| elements is asserted to be ergodic. | |
| -/ | |
| noncomputable section | |
| open Set | |
| open scoped BigOperators | |
| namespace DukePrimeDegree | |
| variable {I : Type*} [Fintype I] [DecidableEq I] | |
| def diagonalParameterGenerators (p : I × Bool) : I → ℝ := | |
| Pi.single p.1 (if p.2 then Real.sqrt 2 else 1) | |
| def singleCoordinateAddHom (i : I) : ℝ →+ (I → ℝ) where | |
| toFun := fun t : ℝ => (Pi.single i t : I → ℝ) | |
| map_zero' := by ext j; simp | |
| map_add' s t := by ext j; by_cases h : i = j <;> simp [h] | |
| omit [Fintype I] in | |
| lemma continuous_singleCoordinateAddHom (i : I) : | |
| Continuous (singleCoordinateAddHom i : ℝ → I → ℝ) := by | |
| apply continuous_pi | |
| intro j | |
| by_cases h : i = j | |
| · subst j | |
| have he : (fun t : ℝ => singleCoordinateAddHom i t i) = id := by | |
| funext t | |
| simp [singleCoordinateAddHom] | |
| change Continuous (fun t : ℝ => singleCoordinateAddHom i t i) | |
| rw [he] | |
| exact continuous_id | |
| · simpa [singleCoordinateAddHom, h] using | |
| (continuous_const : Continuous (fun _ : ℝ => (0 : ℝ))) | |
| theorem dense_diagonalParameterGenerators : | |
| Dense (AddSubgroup.closure (range (diagonalParameterGenerators (I := I))) : Set (I → ℝ)) := by | |
| let H := AddSubgroup.closure (range (diagonalParameterGenerators (I := I))) | |
| have hline (i : I) (t : ℝ) : Pi.single i t ∈ H.topologicalClosure := by | |
| let S : AddSubgroup ℝ := H.topologicalClosure.comap (singleCoordinateAddHom i) | |
| have hSclosed : IsClosed (S : Set ℝ) := | |
| H.isClosed_topologicalClosure.preimage (continuous_singleCoordinateAddHom i) | |
| have h1 : (1 : ℝ) ∈ S := by | |
| apply H.le_topologicalClosure | |
| apply AddSubgroup.subset_closure | |
| exact ⟨(i, false), by simp [diagonalParameterGenerators, singleCoordinateAddHom]⟩ | |
| have h2 : Real.sqrt 2 ∈ S := by | |
| apply H.le_topologicalClosure | |
| apply AddSubgroup.subset_closure | |
| exact ⟨(i, true), by simp [diagonalParameterGenerators, singleCoordinateAddHom]⟩ | |
| have hd : Dense (AddSubgroup.closure {Real.sqrt 2, 1} : Set ℝ) := | |
| dense_addSubgroupClosure_pair_iff.mpr (by simpa using irrational_sqrt_two) | |
| have hle : AddSubgroup.closure {Real.sqrt 2, 1} ≤ S := | |
| (AddSubgroup.closure_le _).mpr (by rw [pair_subset_iff]; exact ⟨h2, h1⟩) | |
| have hdS : Dense (S : Set ℝ) := hd.mono hle | |
| have ht : t ∈ (S : Set ℝ) := by | |
| rw [← hSclosed.closure_eq] | |
| exact hdS t | |
| exact ht | |
| intro v | |
| change v ∈ closure (H : Set (I → ℝ)) | |
| change v ∈ H.topologicalClosure | |
| have he : (∑ i, Pi.single i (v i)) = v := by | |
| ext j | |
| simp | |
| rw [← he] | |
| exact H.topologicalClosure.sum_mem (fun i _ => hline i (v i)) | |
| def multiplicativeParameterGenerators (p : I × Bool) : Multiplicative (I → ℝ) := | |
| Multiplicative.ofAdd (diagonalParameterGenerators p) | |
| lemma dense_multiplicativeParameterGenerators : | |
| Dense (Subgroup.closure (range (multiplicativeParameterGenerators (I := I))) : | |
| Set (Multiplicative (I → ℝ))) := by | |
| have he : (AddSubgroup.closure (range (diagonalParameterGenerators (I := I)))).toSubgroup = | |
| Subgroup.closure (range (multiplicativeParameterGenerators (I := I))) := by | |
| rw [AddSubgroup.toSubgroup_closure] | |
| congr 1 | |
| rw [← he] | |
| exact dense_diagonalParameterGenerators | |
| /-- On an abelian group the domain-action type tag is also a group | |
| homomorphism, so dense generators transfer to the Koopman action. -/ | |
| def parameterDomainHom : Multiplicative (I → ℝ) →* | |
| (Multiplicative (I → ℝ))ᵈᵐᵃ where | |
| toFun := DomMulAct.mk | |
| map_one' := rfl | |
| map_mul' g h := by rw [DomMulAct.mk_mul, mul_comm] | |
| def domainParameterGenerators (p : I × Bool) : (Multiplicative (I → ℝ))ᵈᵐᵃ := | |
| parameterDomainHom (multiplicativeParameterGenerators p) | |
| theorem dense_domainParameterGenerators : | |
| Dense (Subgroup.closure (range (domainParameterGenerators (I := I))) : | |
| Set (Multiplicative (I → ℝ))ᵈᵐᵃ) := by | |
| have he : (Subgroup.closure (range (multiplicativeParameterGenerators (I := I)))).map | |
| parameterDomainHom = Subgroup.closure (range (domainParameterGenerators (I := I))) := by | |
| rw [MonoidHom.map_closure, ← Set.range_comp] | |
| rfl | |
| rw [← he, Subgroup.coe_map] | |
| have hd : DenseRange (parameterDomainHom (I := I)) := DomMulAct.mk.surjective.denseRange | |
| exact hd.dense_image DomMulAct.continuous_mk dense_multiplicativeParameterGenerators | |
| end DukePrimeDegree | |
| end | |
| end OAI | |