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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 | import OAI.MeasureTheory.Falconer.Estimates.CommonRegularFamily
import OAI.MeasureTheory.Falconer.Geometry.ProjectionCells
namespace OAI
/-! Common regular parameters with large subsets having small projections.
This is the exact finite configuration produced before the projection theorem
in Orponen--Shmerkin, Appendix A.7. No projection estimate is assumed here. -/
namespace Falconer
theorem common_regular_small_projections {α : Type*} (J : Finset α) (hJ : J.Nonempty)
(A : α → Finset (Space 2)) (direction : α → ℝ) {δ s K m R : ℝ}
(hδ : 0 ≤ δ) (hK : 0 ≤ K) (hm : 0 < m) (hR : 0 < R)
(hsize : ∀ j ∈ J, m ≤ ((A j).card : ℝ))
(hunion : ((J.biUnion A).card : ℝ) ≤ R*m)
(hA : ∀ j ∈ J, NormalizedFiniteBallBound δ s K (A j))
(M : ℕ) (hproj : ∀ j ∈ J, (projectionCells δ (direction j) (A j)).card ≤ M) :
∃ G ⊆ J.biUnion A, G.Nonempty ∧ m/2 ≤ (G.card : ℝ) ∧
NormalizedFiniteBallBound δ s (4*R^2*K) G ∧
∃ Y ⊆ J, Y.Nonempty ∧ (J.card : ℝ) ≤ 4*R*(Y.card : ℝ) ∧
∀ j ∈ Y, m/4 ≤ ((G ∩ A j).card : ℝ) ∧
(projectionCells δ (direction j) (G ∩ A j)).card ≤ M := by
obtain ⟨G,hGU,hGcard,hGregular,hGmass⟩ := common_regular_family J hJ A hδ hK hm hR
hsize hunion hA
obtain ⟨Y,hYJ,hYcard,hYsize⟩ := popular_family_rows J A G hGU hm hR.le hunion hGmass
have hG : G.Nonempty := Finset.card_pos.mp (by
have hh := (half_pos hm).trans_le hGcard
exact_mod_cast hh)
have hY : Y.Nonempty := by
apply Finset.card_pos.mp
by_contra h
have hz : Y.card = 0 := Nat.eq_zero_of_not_pos h
rw [hz,Nat.cast_zero,mul_zero] at hYcard
have hJpos : 0 < (J.card : ℝ) := Nat.cast_pos.mpr (Finset.card_pos.mpr hJ)
linarith
refine ⟨G,hGU,hG,hGcard,hGregular,Y,hYJ,hY,hYcard,?_⟩
intro j hj
exact ⟨hYsize j hj,
(Finset.card_le_card (projectionCells_mono δ (direction j) Finset.inter_subset_right)).trans
(hproj j (hYJ hj))⟩
end Falconer
end OAI
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