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d4317d7 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 | import OAI.MeasureTheory.Falconer.Geometry.CommonRatioProjections
import OAI.MeasureTheory.Falconer.Geometry.ScalarProjectionDoubling
namespace OAI
/-! Normalized common-set projection estimates in the balanced Cartesian
case, including the doubling needed for coefficient arithmetic. -/
namespace Falconer
open scoped Pointwise
theorem normalize_ratio_product_bound {D C K ρ N A : ℝ}
(hC : 0 ≤ C) (hρ : 0 < ρ) (hN : 0 < N)
(hA : ρ*N ≤ A)
(hbound : D*ρ^12*N*N^2 ≤ C*(K*N)^4) :
D ≤ (C*K^4/ρ^13)*A := by
have hd : D ≤ C*K^4/ρ^12*N := by
calc
_ ≤ C*(K*N)^4/(ρ^12*N*N^2) :=
(le_div_iff₀ (by positivity : 0 < ρ^12*N*N^2)).2
(by simpa only [mul_assoc] using hbound)
_ = _ := by field_simp
have hstep := mul_le_mul_of_nonneg_left hA (by positivity : 0 ≤ C*K^4/ρ^13)
calc
_ ≤ C*K^4/ρ^12*N := hd
_ = (C*K^4/ρ^13)*(ρ*N) := by field_simp
_ ≤ _ := hstep
theorem common_normalized_ratio_projections {α : Type*} {δ ρ K : ℝ}
(hδ : 0 < δ) (hρ : 0 < ρ) (hK : 0 ≤ K)
(I : Finset α) (X Y : Finset ℝ) (A B : α → Finset ℝ) (c : α → ℝ)
(hI : I.Nonempty) (hX : X.Nonempty) (hY : Y.Nonempty) (hXY : X.card = Y.card)
(hXsep : ∀ x ∈ X, ∀ y ∈ X, x ≠ y → δ ≤ |x-y|)
(hYsep : ∀ x ∈ Y, ∀ y ∈ Y, x ≠ y → δ ≤ |x-y|)
(hA : ∀ i ∈ I, A i ⊆ X) (hB : ∀ i ∈ I, B i ⊆ Y)
(hsizeA : ∀ i ∈ I, ρ*(X.card : ℝ) ≤ ((A i).card : ℝ))
(hsizeB : ∀ i ∈ I, ρ*(Y.card : ℝ) ≤ ((B i).card : ℝ))
(M : ℕ) (hc : ∀ i ∈ I, c i ≠ 0 ∧ |c i| ≤ M ∧ |(c i)⁻¹| ≤ M)
(hproj : ∀ i ∈ I, ((scalarCells δ (A i+(B i).image (fun y => c i*y))).card : ℝ) ≤
K*(X.card : ℝ)) :
∃ i₀ ∈ I, ∃ J ⊆ I,
ρ^4*(I.card : ℝ) ≤ 2*(J.card : ℝ) ∧ (A i₀).Nonempty ∧
((scalarCells δ (A i₀+A i₀)).card : ℝ) ≤
(1024*(2*M+3)^2*K^4/ρ^6)*(scalarCells δ (A i₀)).card ∧
∀ j ∈ J,
((scalarCells δ (A i₀-(A i₀).image (fun x => (c j/c i₀)*x))).card : ℝ) ≤
(4096*(2*(M*M)+3)*(2*M+3)^2*K^4/ρ^13)*(scalarCells δ (A i₀)).card := by
classical
obtain ⟨i₀,hi₀,J,hJI,hJ,hraw⟩ := common_small_ratio_projections hδ hρ
(by positivity : 0 ≤ K*(X.card : ℝ)) I X Y A B c hI hX hY hXsep hYsep
hA hB hsizeA hsizeB M hc hproj
have hN : 0 < (X.card : ℝ) := Nat.cast_pos.mpr (Finset.card_pos.mpr hX)
have hAi0 : 0 < ((A i₀).card : ℝ) := (mul_pos hρ hN).trans_le (hsizeA i₀ hi₀)
have hAi : (A i₀).Nonempty := Finset.card_pos.mp (Nat.cast_pos.mp hAi0)
have hAc : (scalarCells δ (A i₀)).card = (A i₀).card :=
Finset.card_image_of_injOn (scalarCells_injective hδ
(fun x hx y hy hxy => hXsep x (hA i₀ hi₀ hx) y (hA i₀ hi₀ hy) hxy))
have hNY : (Y.card : ℝ) = (X.card : ℝ) := by exact_mod_cast hXY.symm
have hbase : ((scalarCells δ (A i₀+(B i₀).image (fun y => c i₀*y))).card : ℝ) ≤
(K/ρ)*(scalarCells δ (A i₀)).card := by
apply (hproj i₀ hi₀).trans
rw [hAc]
have hh := mul_le_mul_of_nonneg_left (hsizeA i₀ hi₀) (div_nonneg hK hρ.le)
calc
_ = (K/ρ)*(ρ*(X.card : ℝ)) := by field_simp
_ ≤ _ := hh
have hBsize : ρ*((scalarCells δ (A i₀)).card : ℝ) ≤ ((B i₀).card : ℝ) := by
rw [hAc]
calc
_ ≤ ρ*(X.card : ℝ) := mul_le_mul_of_nonneg_left
(Nat.cast_le.mpr (Finset.card_le_card (hA i₀ hi₀))) hρ.le
_ = ρ*(Y.card : ℝ) := by rw [hNY]
_ ≤ _ := hsizeB i₀ hi₀
have hd := scalar_doubling_from_projection hδ (hc i₀ hi₀).1 hρ (A i₀) (B i₀)
hAi M (hc i₀ hi₀).2.2
(fun x hx y hy hxy => hYsep x (hB i₀ hi₀ hx) y (hB i₀ hi₀ hy) hxy) hBsize hbase
refine ⟨i₀,hi₀,J,hJI,hJ,hAi,?_,?_⟩
· convert hd using 1
field_simp
· intro j hj
have hb := hraw j hj
rw [hNY] at hb
have hh := normalize_ratio_product_bound
(by positivity : 0 ≤ (4096 : ℝ)*(2*(M*M)+3)*(2*M+3)^2)
hρ hN (hsizeA i₀ hi₀) hb
rw [hAc]
exact hh
end Falconer
end OAI
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