openAI-math-dataset / lean /OAI /MeasureTheory /Falconer /Geometry /CommonNormalizedProjections.lean
| 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 | |