| import OAI.MeasureTheory.Falconer.Estimates.DyadicCenters |
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| namespace OAI |
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| /-! |
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| namespace Falconer |
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| theorem DyadicCube.centers_separated {d n : ℕ} (Q : DyadicCube d) (hQ : 0 < Q.side) |
| {p q : List (Fin d → Bool)} (hp : p.length = n) (hq : q.length = n) (hpq : p ≠ q) : |
| Q.side/(2:ℝ)^n ≤ dist (Q.descendant p).center (Q.descendant q).center := by |
| classical |
| have hcode : dyadicCode p ≠ dyadicCode q := fun h => hpq (dyadicCode_injective_at_depth hp hq h) |
| have hex : ∃ i, dyadicCode p i ≠ dyadicCode q i := by |
| by_contra h |
| push Not at h |
| exact hcode (funext h) |
| obtain ⟨i,hi⟩ := hex |
| have hz : (1:ℤ) ≤ |dyadicCode p i-dyadicCode q i| := by |
| rcases lt_or_gt_of_ne (sub_ne_zero.mpr hi) with h | h |
| · rw [abs_of_neg h]; omega |
| · rw [abs_of_pos h]; omega |
| have hr : (1:ℝ) ≤ |(dyadicCode p i:ℝ)-(dyadicCode q i:ℝ)| := by exact_mod_cast hz |
| have hs : 0 < Q.side/(2:ℝ)^n := by positivity |
| have hcoord : (Q.descendant p).center i-(Q.descendant q).center i = |
| (Q.side/(2:ℝ)^n)*((dyadicCode p i:ℝ)-(dyadicCode q i:ℝ)) := by |
| rw [Q.descendant_center,Q.descendant_center,hp,hq] |
| ring |
| have hb := PiLp.norm_apply_le ((Q.descendant p).center-(Q.descendant q).center) i |
| change |(Q.descendant p).center i-(Q.descendant q).center i| ≤ |
| ‖(Q.descendant p).center-(Q.descendant q).center‖ at hb |
| rw [hcoord,abs_mul,abs_of_pos hs] at hb |
| rw [dist_eq_norm] |
| nlinarith |
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| end Falconer |
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| end OAI |
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