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import OAI.Geometry.Cannon.ScalarLimits.CompactVariationRegion
namespace OAI
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noncomputable section
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namespace CannonRelease
open CannonGraph CannonVisual CannonCells CannonCharts Set Metric
universe v
variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G]
variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ)
local instance : Setoid (BasedRay D) := raySetoid D
theorem boundary_modulus_subexponential (φ : Boundary D ≃ₜ SphereTwo) :
let _ := visualMetricSpace D hδ
let a := Real.exp (2*visualExponent δ)
let LE := 8*Real.exp (visualExponent δ*(4*(8*δ+6):ℕ))
∀ d : ℝ, 0 < d → d < diam (univ : Set (Boundary D)) → d ≤ 1/(LE*3^2) →
∀ (ι : ℕ → Type*) [∀ n, Fintype (ι n)] (p : ∀ n, ι n → Boundary D),
(∀ n i j, i ≠ j → (a^n)⁻¹ ≤ dist (p n i) (p n j)) →
(∀ n x, ∃ i, dist (p n i) x ≤ (a^n)⁻¹) →
∀ lam : ℝ, 1 < lam → ∃ B : ℝ, 0 < B ∧ ∀ n,
modulus (fun i => ball (p n i) (2*(a^n)⁻¹)) (macroFamily d) ≤ B*lam^n := by
classical
let _ := visualMetricSpace D hδ
dsimp only
intro d hd hdia hdc ι hι p hsep hnet lam hlam
have hn := boundary_exponential_records_impossible D hδ φ d hd hdia hdc ι p hsep hnet lam hlam
have hbnd : ∃ B : ℝ, ∀ n,
modulus (fun i => ball (p n i) (2*(Real.exp (2*visualExponent δ)^n)⁻¹))
(macroFamily d)/lam^n ≤ B := by
by_contra h
apply hn
intro B
by_contra hB
apply h
refine ⟨B,fun n => ?_⟩
exact le_of_not_gt (fun hh => hB ⟨n,hh⟩)
obtain ⟨B,hB⟩ := hbnd
refine ⟨max B 1,lt_of_lt_of_le zero_lt_one (le_max_right ..),fun n => ?_⟩
have hh := hB n
have hl : 0 < lam^n := pow_pos (lt_trans zero_lt_one hlam) n
exact (div_le_iff₀ hl).mp (hh.trans (le_max_left ..))
end CannonRelease
end
section
open Set Topology Filter Metric unitInterval
open scoped InnerProductSpace
open Set unitInterval
namespace CannonScalarLocal
open CannonGrid CannonCells CannonRegularization CannonChart
variable {X ι : Type*} [MetricSpace X] [Fintype ι]
local instance : LocallyPathConnectedSpace unitInterval :=
(isQuotientMap_projIcc (a := (0 : ℝ)) (b := 1) (h := zero_le_one)).locallyPathConnectedSpace
local instance : PathConnectedSpace Square :=
pathConnectedSpace_iff_connectedSpace.mpr inferInstance
end CannonScalarLocal
namespace CannonApprox
open CannonGrid CannonRegularization CannonLevel CannonCells CannonBands
variable {ι : Type*} [Fintype ι]
local instance : PathConnectedSpace (Strip height (1 / 3) (2 / 3)) :=
strip_pathConnected _ _ (by norm_num) (by norm_num) (by norm_num)
end CannonApprox
open Set Metric Topology
open Set Topology unitInterval
open Set Filter Topology
open scoped BigOperators
open Set unitInterval
open Set MeasureTheory
open scoped NNReal ENNReal
namespace CannonAtomic
attribute [local instance] Classical.propDecidable
variable {X ι : Type*} [MeasurableSpace X] [Fintype ι]
end CannonAtomic
namespace CannonAtomic
open Set MeasureTheory
attribute [local instance] Classical.propDecidable
end CannonAtomic
open Set Filter Metric Topology
open Set Filter Topology
open Set Filter MeasureTheory Topology
open scoped NNReal ENNReal
namespace CannonScalarLimit
open CannonAtomic CannonAnnularCompactness CannonScales Set Filter Metric Topology MeasureTheory
attribute [local instance] Classical.propDecidable
end CannonScalarLimit
namespace CannonScalarLimit
open CannonAtomic CannonAnnularCompactness Set Filter Metric Topology MeasureTheory
open scoped NNReal ENNReal
attribute [local instance] Classical.propDecidable
end CannonScalarLimit
open Set Filter Topology TopologicalSpace Metric
end
end
end
end
end
end
end
end
end OAI
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