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7f37411 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 | 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
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