File size: 3,091 Bytes
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 | 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
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