openAI-math-dataset / lean /OAI /NumberTheory /DukePrimeDegree /Geometry /ControlledDistanceCuts.lean
| import OAI.NumberTheory.DukePrimeDegree.Geometry.QuantitativeDistanceNet | |
| import OAI.NumberTheory.DukePrimeDegree.Geometry.DyadicBoundaryCuts | |
| namespace OAI | |
| /-! # Small quotient observations with controlled distance boundaries | |
| Each logarithmic ball admits finitely many distance cuts that cover it, | |
| have small relative lifts, and satisfy a common dyadic boundary estimate. | |
| Both the number of cuts and their boundary constants are explicit. | |
| -/ | |
| noncomputable section | |
| open Set MeasureTheory | |
| open scoped ENNReal | |
| namespace DukePrimeDegree | |
| open PrimeDegreePackets | |
| theorem exists_controlled_distance_cuts {n : ℕ} (hn : 0 < n) | |
| (μ : Measure (X n)) [IsProbabilityMeasure μ] | |
| (R r : ℝ) (hr : 0 < r) (hr1 : r ≤ 1) : | |
| let τ := Real.log (1+r) | |
| ∃ (N : ℕ) (z : Fin N → X n) (c : Fin N → ℝ), | |
| (N:ℝ) ≤ quotientNetBound n R (τ/4) ∧ | |
| (∀ i, τ/3 < c i ∧ c i < τ/2 ∧ | |
| μ {x | quotientLogDistance (z i) x=c i}=0 ∧ | |
| ∀ k : ℕ, μ (realBoundaryStrip (quotientLogDistance (z i)) (c i) ((1/4:ℝ)^k)) ≤ | |
| ENNReal.ofReal (48/τ*(1/2:ℝ)^k)) ∧ | |
| (∀ x, quotientLogDistance (Quotient.mk _ (1 : G n)) x ≤ R → | |
| ∃ i, quotientLogDistance (z i) x < c i) ∧ | |
| ∀ i, ∀ x, quotientLogDistance (z i) x < c i → | |
| ∀ y, quotientLogDistance (z i) y < c i → | |
| ∃ g ∈ matrixBall n r, rightAction x g=y := by | |
| classical | |
| let τ := Real.log (1+r) | |
| have hτ : 0 < τ := Real.log_pos (by linarith) | |
| have hτr : τ ≤ r := by | |
| have H := Real.log_le_sub_one_of_pos (show 0 < 1+r by linarith) | |
| dsimp [τ] | |
| linarith | |
| obtain ⟨N,z,hN,hcover⟩ := exists_quantitative_distance_net hn R (τ/4) | |
| (by positivity) (by linarith) | |
| have hex (i : Fin N) := exists_dyadic_boundary_cut μ | |
| (continuous_quotientLogDistance n (z i)).measurable | |
| (show τ/3 < τ/2 by linarith) | |
| choose c hc hnull hboundary using hex | |
| refine ⟨N,z,c,hN,?_,?_,?_⟩ | |
| · intro i | |
| refine ⟨(hc i).1,(hc i).2,hnull i,?_⟩ | |
| intro k | |
| convert hboundary i k using 1 | |
| congr 1 | |
| change 48/τ*(1/2:ℝ)^k = 8/(τ/2-τ/3)*(1/2:ℝ)^k | |
| have hne := hτ.ne' | |
| field_simp [hne] | |
| ring | |
| · intro x hx | |
| obtain ⟨i,hi⟩ := hcover x hx | |
| exact ⟨i,hi.trans (by linarith [(hc i).1])⟩ | |
| · intro i x hx y hy | |
| apply exists_matrix_lift_of_logDistance_lt hr | |
| have H := quotientLogDistance_triangle x (z i) y | |
| rw [quotientLogDistance_symm x (z i)] at H | |
| change quotientLogDistance x y < τ | |
| linarith [(hc i).2] | |
| end DukePrimeDegree | |
| end | |
| end OAI | |