| import OAI.MeasureTheory.Falconer.Estimates.FiniteFamilyOverlap |
| import Mathlib.Data.Finset.Prod |
|
|
| namespace OAI |
|
|
| /-! Simultaneous large intersections for both factors of a family of |
| rectangles. The reference rectangle is an actual member of the family. -/ |
| namespace Falconer |
|
|
| theorem common_rectangular_subsets {α : Type*} |
| (I : Finset α) (X Y : Finset ℝ) (A B : α → Finset ℝ) |
| (hI : I.Nonempty) (hX : X.Nonempty) (hY : Y.Nonempty) |
| {ρ : ℝ} (hρ : 0 < ρ) |
| (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)) : |
| ∃ i₀ ∈ I, ∃ J ⊆ I, |
| ρ^4*(I.card : ℝ) ≤ 2*(J.card : ℝ) ∧ |
| ∀ j ∈ J, |
| ρ^4*(X.card : ℝ)/2 ≤ ((A i₀ ∩ A j).card : ℝ) ∧ |
| ρ^4*(Y.card : ℝ)/2 ≤ ((B i₀ ∩ B j).card : ℝ) := by |
| classical |
| have hXpos : 0 < (X.card : ℝ) := Nat.cast_pos.mpr (Finset.card_pos.mpr hX) |
| have hYpos : 0 < (Y.card : ℝ) := Nat.cast_pos.mpr (Finset.card_pos.mpr hY) |
| let R := fun i => (A i) ×ˢ (B i) |
| have hRsub (i) (hi : i ∈ I) : R i ⊆ X ×ˢ Y := by |
| intro p hp |
| obtain ⟨ha,hb⟩ := Finset.mem_product.mp hp |
| exact Finset.mem_product.mpr ⟨hA i hi ha,hB i hi hb⟩ |
| have hRsize (i) (hi : i ∈ I) : ρ^2*(X.card : ℝ)*(Y.card : ℝ) ≤ ((R i).card : ℝ) := by |
| have hh := mul_le_mul (hsizeA i hi) (hsizeB i hi) |
| (mul_pos hρ hYpos).le (Nat.cast_nonneg _) |
| simpa only [R,Finset.card_product,Nat.cast_mul,pow_two,mul_assoc,mul_left_comm,mul_comm] using hh |
| obtain ⟨i₀,hi₀,J,hJI,hJ,hrect⟩ := finite_family_common_row I (X ×ˢ Y) R |
| hI (hX.product hY) (by positivity : 0 < ρ^2*(X.card : ℝ)*(Y.card : ℝ)) hRsub hRsize |
| have hn : (ρ^2*(X.card : ℝ)*(Y.card : ℝ))^2/((X.card : ℝ)*(Y.card : ℝ)) = |
| ρ^4*(X.card : ℝ)*(Y.card : ℝ) := by |
| field_simp |
| have hnorm : (ρ^2*(X.card : ℝ)*(Y.card : ℝ))^2/(2*((X.card : ℝ)*(Y.card : ℝ))) = |
| ρ^4*(X.card : ℝ)*(Y.card : ℝ)/2 := by |
| field_simp |
| simp only [Finset.card_product,Nat.cast_mul,hn] at hJ |
| refine ⟨i₀,hi₀,J,hJI,?_,?_⟩ |
| · apply (mul_le_mul_iff_left₀ (mul_pos hXpos hYpos)).1 |
| convert hJ using 1 <;> ring |
| · intro j hj |
| have hh := hrect j hj |
| simp only [R,Finset.product_inter_product,Finset.card_product,Nat.cast_mul,hnorm] at hh |
| have ha : ((A i₀ ∩ A j).card : ℝ) ≤ (X.card : ℝ) := |
| Nat.cast_le.mpr (Finset.card_le_card (Finset.inter_subset_left.trans (hA i₀ hi₀))) |
| have hb : ((B i₀ ∩ B j).card : ℝ) ≤ (Y.card : ℝ) := |
| Nat.cast_le.mpr (Finset.card_le_card (Finset.inter_subset_left.trans (hB i₀ hi₀))) |
| constructor |
| · have hx := hh.trans (mul_le_mul_of_nonneg_left hb (Nat.cast_nonneg _)) |
| apply (mul_le_mul_iff_left₀ hYpos).1 |
| convert hx using 1 |
| ring |
| · have hy := hh.trans (mul_le_mul_of_nonneg_right ha (Nat.cast_nonneg _)) |
| apply (mul_le_mul_iff_left₀ hXpos).1 |
| convert hy using 1 <;> ring |
|
|
| end Falconer |
|
|
| end OAI |
|
|
