| import OAI.MeasureTheory.Falconer.Recursion.FrostmanProfiles |
| import OAI.MeasureTheory.Falconer.Estimates.RegularizationSubpower |
| import OAI.MeasureTheory.Falconer.Shells.ShellSummability |
|
|
| namespace OAI |
|
|
| /-! # Polynomial restoration losses and the coarse product-filter cost -/ |
|
|
| namespace Falconer |
| open Filter |
|
|
| theorem eventual_polynomial_decay {a : ℝ} (ha : 0 < a) (C : ℝ) (k : ℕ) : |
| ∃ N₀ : ℕ, ∀ N ≥ N₀, |
| C*((N:ℝ)+1)^k*((2:ℝ)^N)^(-a) ≤ ((2:ℝ)^N)^(-a/2) := by |
| obtain ⟨N₁,h₁⟩ := Regularization.fixed_constant_absorbed (by positivity : 0 < a/4) C |
| obtain ⟨N₂,h₂⟩ := PlanarFalconer.fixed_polynomial_subpower k (by positivity : 0 < a/4) |
| refine ⟨max N₁ N₂,?_⟩ |
| intro N hN |
| have hR : 0 < (2:ℝ)^N := by positivity |
| have hp : ((N:ℝ)+1)^k ≤ ((2:ℝ)^N)^(a/4) := |
| (pow_le_pow_left₀ (by positivity) |
| (by nlinarith [show (0:ℝ) ≤ (N:ℝ) from Nat.cast_nonneg N] : (N:ℝ)+1 ≤ 4*N+1) k).trans |
| (h₂ N ((le_max_right _ _).trans hN)) |
| calc |
| _ ≤ (((2:ℝ)^N)^(a/4)*((2:ℝ)^N)^(a/4))*((2:ℝ)^N)^(-a) := |
| mul_le_mul_of_nonneg_right |
| (mul_le_mul (h₁ N ((le_max_left _ _).trans hN)) hp (by positivity) (by positivity)) (by positivity) |
| _ = _ := by rw [← Real.rpow_add hR,← Real.rpow_add hR]; congr 1; ring |
|
|
| theorem eventual_partition_shell_decay {β θ : ℝ} (hβ : 0 < β) (hθ : 2*θ ≤ β/64) : |
| ∃ N₀ : ℕ, ∀ N ≥ N₀, ∀ k : ℕ, (k:ℝ) ≤ ((2:ℝ)^N)^θ → |
| 2*(k:ℝ)^2*((2:ℝ)^N)^(-β/16) ≤ ((2:ℝ)^N)^(-β/32) := by |
| obtain ⟨N₀,h₀⟩ := Regularization.fixed_constant_absorbed (by positivity : 0 < β/64) 2 |
| refine ⟨N₀,?_⟩ |
| intro N hN k hk |
| have hR : 0 < (2:ℝ)^N := by positivity |
| have hR1 : 1 ≤ (2:ℝ)^N := one_le_pow₀ (by norm_num) |
| have hk2 : (k:ℝ)^2 ≤ ((2:ℝ)^N)^(β/64) := by |
| calc |
| _ ≤ (((2:ℝ)^N)^θ)^2 := pow_le_pow_left₀ (Nat.cast_nonneg _) hk 2 |
| _ = ((2:ℝ)^N)^(2*θ) := by |
| rw [← Real.rpow_natCast,← Real.rpow_mul hR.le] |
| congr 1 |
| ring |
| _ ≤ _ := Real.rpow_le_rpow_of_exponent_le hR1 hθ |
| calc |
| _ ≤ (((2:ℝ)^N)^(β/64)*((2:ℝ)^N)^(β/64))*((2:ℝ)^N)^(-β/16) := |
| mul_le_mul_of_nonneg_right (mul_le_mul (h₀ N hN) hk2 (by positivity) (by positivity)) (by positivity) |
| _ = _ := by rw [← Real.rpow_add hR,← Real.rpow_add hR]; congr 1; ring |
|
|
| theorem dyadic_scale_power (N : ℕ) (a : ℝ) : ((2:ℝ)^N)^a = (2:ℝ)^(a*(N:ℝ)) := by |
| rw [← Real.rpow_natCast,← Real.rpow_mul (by norm_num : (0:ℝ) ≤ 2),mul_comm] |
|
|
| theorem summable_scale_decay {a : ℝ} (ha : 0 < a) : |
| Summable (fun N : ℕ => ((2:ℝ)^N)^(-a)) := by |
| simp_rw [dyadic_scale_power] |
| exact PlanarFalconer.summable_dyadic_decay ha |
|
|
| theorem summable_polynomial_scale_decay {a : ℝ} (ha : 0 < a) (C : ℝ) (k : ℕ) : |
| Summable (fun N : ℕ => C*((N:ℝ)+1)^k*((2:ℝ)^N)^(-a)) := by |
| obtain ⟨N₀,h₀⟩ := eventual_polynomial_decay (half_pos ha) |C| k |
| apply (summable_scale_decay (show 0 < a/4 by positivity)).of_norm_bounded_eventually_nat |
| filter_upwards [eventually_ge_atTop N₀] with N hN |
| rw [norm_mul,norm_mul,Real.norm_eq_abs,Real.norm_eq_abs,Real.norm_eq_abs, |
| abs_of_nonneg (by positivity : 0 ≤ ((N:ℝ)+1)^k), |
| abs_of_nonneg (Real.rpow_nonneg (by positivity) _)] |
| have hh := h₀ N hN |
| have hR1 : 1 ≤ (2:ℝ)^N := one_le_pow₀ (by norm_num) |
| calc |
| _ ≤ |C| *((N:ℝ)+1)^k*((2:ℝ)^N)^(-(a/2)) := |
| mul_le_mul_of_nonneg_left (Real.rpow_le_rpow_of_exponent_le hR1 (by linarith)) (by positivity) |
| _ ≤ _ := by |
| have he : -(a/2)/2 = -(a/4) := by ring |
| simpa only [he] using hh |
|
|
| end Falconer |
|
|
| end OAI |
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|
