| import Mathlib |
| import OAI.MeasureTheory.FalconerCompletion.Sphere.SphereMeasure |
|
|
| namespace OAI |
|
|
| /-! Quantitative estimates for the distance-set construction. -/ |
|
|
| section |
| section |
| namespace CurrentFalconer.TreeND |
| abbrev CellPath (d : ℕ) := List (Fin d → Bool) |
| variable {d : ℕ} |
|
|
| def dyadicPaths {d : ℕ} : ℕ → Finset (CellPath d) |
| | 0 => {[]} |
| | n+1 => Finset.univ.biUnion fun a : Fin d → Bool => |
| (dyadicPaths n).image (List.cons a) |
|
|
| @[simp] theorem mem_dyadicPaths (n : ℕ) (p : CellPath d) : |
| p ∈ dyadicPaths n ↔ p.length = n := by |
| induction n generalizing p with |
| | zero => simp [dyadicPaths] |
| | succ n ih => |
| cases p with |
| | nil => simp [dyadicPaths] |
| | cons a p => simp [dyadicPaths, ih] |
|
|
| lemma dyadicPaths_card_le (N : ℕ) : (dyadicPaths (d := d) N).card ≤ (2^d)^N := by |
| classical |
| induction N with |
| | zero => simp [dyadicPaths] |
| | succ N ih => |
| calc |
| _ ≤ ∑ a ∈ (Finset.univ : Finset (Fin d → Bool)), |
| ((dyadicPaths N).image (List.cons a)).card := Finset.card_biUnion_le |
| _ ≤ ∑ _a ∈ (Finset.univ : Finset (Fin d → Bool)), (2^d)^N := |
| Finset.sum_le_sum (fun _ _ => Finset.card_image_le.trans ih) |
| _ = _ := by simp [pow_succ,Nat.mul_comm] |
|
|
| end CurrentFalconer.TreeND |
| end |
| end |
|
|
| end OAI |
|
|
