| import OAI.MeasureTheory.Falconer.Graphs.DenseGraphParameters |
|
|
| namespace OAI |
|
|
| /-! Dense original graphs with small restricted projections produce the |
| refined family forbidden by the scalar projection theorem. -/ |
| namespace Falconer |
| open scoped Pointwise |
| attribute [local instance] Classical.propDecidable |
|
|
| theorem dense_graph_obstruction {κ : ℝ} (hκ : 0 < κ) (hκ1 : κ ≤ 1) |
| (M : ℕ) (hM : 1 ≤ M) : |
| ∃ γ : ℝ, ∃ N0 : ℕ, 0 < γ ∧ 0 < N0 ∧ ∀ N : ℕ, N0 ≤ N → |
| let δ := (1/2:ℝ)^N |
| let S := Real.exp (γ*(N:ℝ)*Real.log 2) |
| ∀ (t : ℝ) (I X Y : Finset ℝ) (e : ℝ → ℝ → ℝ → Prop) (D : ℝ → Finset ℤ) |
| (G : Finset (Space 2)), |
| 2*κ ≤ t → t ≤ 2 → I.Nonempty → X.Nonempty → Y.Nonempty → X.card = Y.card → |
| (∀ x ∈ X, |x| ≤ 1) → |
| (∀ x ∈ X, ∀ y ∈ X, x ≠ y → δ ≤ |x-y|) → |
| (∀ x ∈ Y, ∀ y ∈ Y, x ≠ y → δ ≤ |x-y|) → |
| (∀ i ∈ I, ∀ x ∈ Y.image (fun y => i*y), ∀ y ∈ Y.image (fun y => i*y), x ≠ y → δ ≤ |x-y|) → |
| (∀ i ∈ I, i ≠ 0 ∧ |i| ≤ M ∧ |i⁻¹| ≤ M) → |
| (∀ i ∈ I, S⁻¹*(X.card:ℝ)*(Y.card:ℝ) ≤ ∑ x ∈ X, ((Y.filter fun y => e i x y).card:ℝ)) → |
| (∀ i ∈ I, ∀ x ∈ X, ∀ y ∈ Y, e i x y → ⌊(x+i*y)/δ⌋ ∈ D i) → |
| (∀ i ∈ I, ((D i).card:ℝ) ≤ S*(X.card:ℝ)) → |
| NormalizedFiniteBallBound δ t S G → |
| (∀ i ∈ I, ∀ x ∈ X, ∀ y ∈ Y, e i x y → rowPoint x y ∈ G) → |
| (G.card:ℝ) ≤ S*(X.card:ℝ)*(Y.card:ℝ) → |
| (∀ u r : ℝ, δ ≤ r → r ≤ 1 → scalarBallMass I u r ≤ S*r^(2*κ)*(I.card:ℝ)) → |
| Real.log (X.card:ℝ) ≤ (1-κ)*(N:ℝ)*Real.log 2 → False := by |
| classical |
| obtain ⟨θ,N0,hθ,hN0,hweak⟩ := weak_graph_obstruction hκ hκ1 M hM |
| obtain ⟨γ,N1,hγ,hN1,hparameters⟩ := dense_graph_parameters hθ |
| refine ⟨γ,max N0 N1,hγ,lt_of_lt_of_le hN0 (le_max_left _ _),?_⟩ |
| intro N hN |
| dsimp only |
| intro t I X Y e D G ht ht2 hI hX hY hXY hXbound hXsep hYsep hcYsep hc hdense hlabels |
| hDcard hreg hgraph hGsize hIball hXsize |
| let δ := (1/2:ℝ)^N |
| let S := Real.exp (γ*(N:ℝ)*Real.log 2) |
| let R := Real.exp (θ*(N:ℝ)*Real.log 2) |
| have hδ : 0 < δ := by positivity |
| obtain ⟨hS,hSR,hprojcost,hdegreecost⟩ := hparameters N ((le_max_right _ _).trans hN) |
| have hex : ∀ i : ℝ, ∃ A B : Finset ℝ, i ∈ I → |
| A ⊆ X ∧ B ⊆ Y ∧ R⁻¹*(X.card:ℝ) ≤ (A.card:ℝ) ∧ |
| R⁻¹*(Y.card:ℝ) ≤ (B.card:ℝ) ∧ |
| ((scalarCells δ (A+B.image (fun y => i*y))).card:ℝ) ≤ R*(X.card:ℝ) ∧ |
| ∀ x ∈ A, R⁻¹*(B.card:ℝ) ≤ ((B.filter fun y => e i x y).card:ℝ) := by |
| intro i |
| by_cases hi : i ∈ I |
| · obtain ⟨A,hAX,B,hBY,hA,hB,hproj,hdegree⟩ := scalar_dilated_weak_refinement |
| hδ hS (hc i hi).1 X Y hX hY hXY hXsep (hcYsep i hi) (e i) (D i) |
| (hdense i hi) (hlabels i hi) (hDcard i hi) |
| refine ⟨A,B,fun _ => ⟨hAX,hBY,?_,?_,?_,?_⟩⟩ |
| · exact (mul_le_mul_of_nonneg_right hdegreecost (Nat.cast_nonneg _)).trans hA |
| · exact (mul_le_mul_of_nonneg_right hdegreecost (Nat.cast_nonneg _)).trans hB |
| · exact hproj.trans (mul_le_mul_of_nonneg_right hprojcost (Nat.cast_nonneg _)) |
| · intro x hx |
| exact (mul_le_mul_of_nonneg_right hdegreecost (Nat.cast_nonneg _)).trans (hdegree x hx) |
| · exact ⟨∅,∅,fun h => (hi h).elim⟩ |
| choose A B hAB using hex |
| have hRreg : NormalizedFiniteBallBound δ t R G := by |
| intro z r hr hr1 |
| have hh := mul_le_mul_of_nonneg_right hSR |
| (mul_nonneg (Real.rpow_nonneg (hδ.le.trans hr) t) (Nat.cast_nonneg G.card)) |
| exact (hreg z r hr hr1).trans (by nlinarith) |
| have hRball : ∀ u r : ℝ, δ ≤ r → r ≤ 1 → scalarBallMass I u r ≤ R*r^(2*κ)*(I.card:ℝ) := by |
| intro u r hr hr1 |
| have hh := mul_le_mul_of_nonneg_right hSR |
| (mul_nonneg (Real.rpow_nonneg (hδ.le.trans hr) (2*κ)) (Nat.cast_nonneg I.card)) |
| exact (hIball u r hr hr1).trans (by nlinarith) |
| apply hweak N ((le_max_left _ _).trans hN) t I X Y A B ht ht2 hI hX hY hXY |
| hXbound hXsep hYsep |
| (fun i hi => (hAB i hi).1) (fun i hi => (hAB i hi).2.1) |
| (fun i hi => (hAB i hi).2.2.1) (fun i hi => (hAB i hi).2.2.2.1) hc |
| (fun i hi => (hAB i hi).2.2.2.2.1) e G hRreg |
| (fun i hi => (hAB i hi).2.2.2.2.2) |
| (fun i hi x hx y hy he => hgraph i hi x ((hAB i hi).1 hx) y ((hAB i hi).2.1 hy) he) |
| (hGsize.trans (mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hSR |
| (Nat.cast_nonneg X.card)) (Nat.cast_nonneg Y.card))) hRball hXsize |
|
|
| end Falconer |
|
|
| end OAI |
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