| import OAI.MeasureTheory.Falconer.Estimates.DyadicCenters |
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| namespace OAI |
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| /-! # A ball meets a bounded number of dyadic cubes of comparable side -/ |
| namespace Falconer |
| open Set Metric |
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| noncomputable def cubeBallHits {d : ℕ} (Q : DyadicCube d) (n : ℕ) |
| (x : Space d) (r : ℝ) : Finset (List (Fin d → Bool)) := by |
| classical |
| exact (cubePaths d n).filter fun p => ((Q.descendant p).carrier ∩ ball x r).Nonempty |
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| theorem mem_cubeBallHits {d : ℕ} (Q : DyadicCube d) (n : ℕ) |
| (x : Space d) (r : ℝ) (p : List (Fin d → Bool)) : |
| p ∈ cubeBallHits Q n x r ↔ p.length = n ∧ |
| ((Q.descendant p).carrier ∩ ball x r).Nonempty := by |
| classical |
| simp [cubeBallHits] |
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| theorem cubeBallHits_coordinate_bound {d n : ℕ} (Q : DyadicCube d) |
| (hQ : 0 < Q.side) (x : Space d) {r : ℝ} |
| (hr : r ≤ Q.side/(2:ℝ)^n) {p q : List (Fin d → Bool)} |
| (hp : p ∈ cubeBallHits Q n x r) (hq : q ∈ cubeBallHits Q n x r) (i : Fin d) : |
| |dyadicCode p i - dyadicCode q i| ≤ (3 : ℤ) := by |
| obtain ⟨hp, u, hu, hux⟩ := (mem_cubeBallHits Q n x r p).mp hp |
| obtain ⟨hq, v, hv, hvx⟩ := (mem_cubeBallHits Q n x r q).mp hq |
| have hui := hu i |
| have hvi := hv i |
| rw [Q.descendant_origin,Q.descendant_side,hp] at hui |
| rw [Q.descendant_origin,Q.descendant_side,hq] at hvi |
| have huv : |u i-v i| ≤ 2*r := by |
| have hb : dist u v < 2*r := |
| (dist_triangle u x v).trans_lt (by |
| rw [dist_comm x v] |
| linarith [mem_ball.mp hux,mem_ball.mp hvx]) |
| exact (PiLp.norm_apply_le (u-v) i).trans (by simpa only [dist_eq_norm] using hb.le) |
| have ht : 0 < Q.side/(2:ℝ)^n := by positivity |
| have hh : |(dyadicCode p i:ℝ)-(dyadicCode q i:ℝ)| ≤ 3 := by |
| rw [abs_le] at huv ⊢ |
| constructor <;> nlinarith |
| exact_mod_cast hh |
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| theorem cubeBallHits_card {d : ℕ} (Q : DyadicCube d) (hQ : 0 < Q.side) |
| (n : ℕ) (x : Space d) {r : ℝ} (hr : r ≤ Q.side/(2:ℝ)^n) : |
| (cubeBallHits Q n x r).card ≤ 7^d := by |
| classical |
| let S := cubeBallHits Q n x r |
| change S.card ≤ 7^d |
| by_cases hS : S.Nonempty |
| · obtain ⟨p₀,hp₀⟩ := hS |
| have hinj : Set.InjOn dyadicCode (S : Set (List (Fin d → Bool))) := by |
| intro p hp q hq hpq |
| exact dyadicCode_injective_at_depth |
| ((mem_cubeBallHits Q n x r p).mp hp).1 |
| ((mem_cubeBallHits Q n x r q).mp hq).1 hpq |
| have hc : (S.image dyadicCode).card = S.card := Finset.card_image_iff.mpr hinj |
| rw [← hc] |
| exact grid_box_card (L := 3) (S.image dyadicCode) (dyadicCode p₀) (by |
| intro z hz i |
| obtain ⟨p,hp,rfl⟩ := Finset.mem_image.mp hz |
| exact cubeBallHits_coordinate_bound Q hQ x hr hp hp₀ i) |
| · simp only [Finset.not_nonempty_iff_eq_empty.mp hS,Finset.card_empty] |
| exact Nat.zero_le _ |
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| end Falconer |
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| end OAI |
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