| import OAI.MeasureTheory.Falconer.Shells.HighShellNormalization |
| import OAI.MeasureTheory.Falconer.Shells.ShellMultipliers |
|
|
| namespace OAI |
|
|
| namespace PlanarFalconer |
| open Set MeasureTheory |
|
|
| theorem high_shell_radial_inverse {v r : ℝ} (hr : (2:ℝ)^(v-1) ≤ r) (j : ℕ) : |
| ‖((((r/(2:ℝ)^v)⁻¹)^j:ℝ):ℂ)‖ ≤ (2:ℝ)^j := by |
| have hp : 0 < r := (Real.rpow_pos_of_pos (by norm_num) _).trans_le hr |
| have he : (2:ℝ)^(v-1) = (2:ℝ)^v/2 := by rw [Real.rpow_sub (by norm_num)]; norm_num |
| have hi : (r/(2:ℝ)^v)⁻¹ ≤ 2 := by |
| rw [inv_div,div_le_iff₀ hp] |
| rw [he] at hr |
| linarith |
| rw [Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (by positivity)] |
| exact pow_le_pow_left₀ (by positivity) hi j |
|
|
| theorem high_shell_radial_sqrt {a v r : ℝ} |
| (hr : r ∈ Icc ((2:ℝ)^(v-1)) ((2:ℝ)^(v+1))) : |
| ‖(((r*(2:ℝ)^(-a))^(1/2:ℝ):ℝ):ℂ)‖ ≤ Real.sqrt (2*(2:ℝ)^(v-a)) := by |
| have hp : 0 ≤ r := (Real.rpow_pos_of_pos (by norm_num) _).le.trans hr.1 |
| have he : (2:ℝ)^(v+1)*(2:ℝ)^(-a) = 2*(2:ℝ)^(v-a) := by |
| rw [Real.rpow_add (by norm_num),Real.rpow_one] |
| have hh : (2:ℝ)^v*(2:ℝ)^(-a) = (2:ℝ)^(v-a) := by |
| rw [← Real.rpow_add (by norm_num)]; rfl |
| calc |
| _ = 2*((2:ℝ)^v*(2:ℝ)^(-a)) := by ring |
| _ = _ := by rw [hh] |
| rw [Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (by positivity),← Real.sqrt_eq_rpow] |
| exact Real.sqrt_le_sqrt ((mul_le_mul_of_nonneg_right hr.2 (by positivity)).trans_eq he) |
|
|
| end PlanarFalconer |
|
|
| end OAI |
|
|
