| 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 |
|
|
