| import Mathlib.Algebra.Order.BigOperators.Group.Finset |
| import Mathlib.Basic.Real.Basic |
| import Mathlib.Tactic.Linarith |
| import Mathlib.Tactic.Positivity |
|
|
| namespace OAI |
|
|
| namespace PlanarFalconer |
|
|
| theorem inverse_distance_dyadic {d : ℝ} (hd : 0 < d) (g k : ℕ) |
| (hmin : 1/(2:ℝ)^(g+k) ≤ d) : |
| 1/d ≤ (2:ℝ)^g + ∑ j ∈ Finset.range k, |
| if d < 1/(2:ℝ)^(g+j) then (2:ℝ)^(g+j+1) else 0 := by |
| induction k with |
| | zero => |
| simp only [Finset.range_zero,Finset.sum_empty,add_zero] at * |
| apply (div_le_iff₀ hd).mpr |
| have hh := (div_le_iff₀ (by positivity : 0 < (2:ℝ)^g)).mp hmin |
| nlinarith |
| | succ k ih => |
| rw [Finset.sum_range_succ] |
| by_cases hcoarse : 1/(2:ℝ)^(g+k) ≤ d |
| · have hh := ih hcoarse |
| have hnon : 0 ≤ (if d < 1/(2:ℝ)^(g+k) then (2:ℝ)^(g+k+1) else 0) := by |
| split_ifs <;> positivity |
| linarith |
| · rw [ite_eq_left (lt_of_not_ge hcoarse)] |
| have hfine : 1/d ≤ (2:ℝ)^(g+k+1) := by |
| apply (div_le_iff₀ hd).mpr |
| have hh := (div_le_iff₀ (by positivity : 0 < (2:ℝ)^(g+(k+1)))).mp hmin |
| simpa only [Nat.add_assoc,mul_comm] using hh |
| have hsum : 0 ≤ ∑ j ∈ Finset.range k, |
| if d < 1/(2:ℝ)^(g+j) then (2:ℝ)^(g+j+1) else 0 := by |
| apply Finset.sum_nonneg |
| intro j _ |
| split_ifs <;> positivity |
| have hg : 0 < (2:ℝ)^g := by positivity |
| linarith |
|
|
| theorem sum_inverse_distance_dyadic {ι : Type*} (S : Finset ι) (d : ι → ℝ) |
| (g k : ℕ) (hd : ∀ i ∈ S, 0 < d i) (hmin : ∀ i ∈ S, 1/(2:ℝ)^(g+k) ≤ d i) : |
| ∑ i ∈ S, 1/d i ≤ (S.card:ℝ)*(2:ℝ)^g + |
| ∑ j ∈ Finset.range k, ((S.filter fun i => d i < 1/(2:ℝ)^(g+j)).card:ℝ)*(2:ℝ)^(g+j+1) := by |
| classical |
| calc |
| _ ≤ ∑ i ∈ S, ((2:ℝ)^g + ∑ j ∈ Finset.range k, |
| if d i < 1/(2:ℝ)^(g+j) then (2:ℝ)^(g+j+1) else 0) := |
| Finset.sum_le_sum (fun i hi => inverse_distance_dyadic (hd i hi) g k (hmin i hi)) |
| _ = _ := by |
| rw [Finset.sum_add_distrib,Finset.sum_comm] |
| simp only [Finset.sum_const,nsmul_eq_mul] |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro j _ |
| rw [← Finset.sum_filter] |
| simp |
|
|
| end PlanarFalconer |
|
|
| end OAI |
|
|
