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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 38 39 40 41 42 43 44 45 46 47 48 49 50 | import OAI.MeasureTheory.Falconer.Shells.HighShellRadialBounds
namespace OAI
namespace PlanarFalconer
open Set MeasureTheory
theorem high_shell_error_power (N a : ℕ) (hN : 1 ≤ N) {v : ℝ} (hv : v ≤ N) (T : ℝ) :
(2:ℝ)^(-(a:ℝ)) * (2*(((2:ℝ)^N)^(-((T+3)/2)))^2 *
(volume.restrict (Icc ((2:ℝ)^(v-1)) ((2:ℝ)^(v+1)))).real univ) ≤
((2:ℝ)^N)^(-T) := by
let R : ℝ := (2:ℝ)^N
have hR : 0 < R := by dsimp [R]; positivity
have hR2 : 2 ≤ R := by
dsimp [R]
exact_mod_cast (show 2^1 ≤ 2^N from Nat.pow_le_pow_right (by omega) hN)
have houter : (2:ℝ)^(-(a:ℝ)) ≤ 1 := by
simpa using Real.rpow_le_rpow_of_exponent_le (by norm_num : (1:ℝ) ≤ 2)
(show -(a:ℝ) ≤ 0 from neg_nonpos.mpr (Nat.cast_nonneg a))
have hlength : (volume.restrict (Icc ((2:ℝ)^(v-1)) ((2:ℝ)^(v+1)))).real univ ≤ 2*R := by
rw [measureReal_def,Measure.restrict_apply_univ,Real.volume_Icc,
ENNReal.toReal_ofReal (sub_nonneg.mpr (Real.rpow_le_rpow_of_exponent_le (by norm_num) (by linarith)))]
have hp : (2:ℝ)^(v+1) ≤ 2*R := by
rw [Real.rpow_add (by norm_num),Real.rpow_one,mul_comm]
apply mul_le_mul_of_nonneg_left _ (by norm_num)
simpa only [Real.rpow_natCast] using Real.rpow_le_rpow_of_exponent_le (by norm_num : (1:ℝ) ≤ 2) hv
linarith [Real.rpow_nonneg (by norm_num : (0:ℝ) ≤ 2) (v-1)]
have he : (R^(-((T+3)/2)))^2 = R^(-(T+3)) := by
rw [← Real.rpow_mul_natCast hR.le]
congr 1
norm_num
change (2:ℝ)^(-(a:ℝ))*(2*(R^(-((T+3)/2)))^2*_) ≤ R^(-T)
rw [he]
calc
_ ≤ 1*(2*R^(-(T+3))*(2*R)) :=
mul_le_mul houter (mul_le_mul_of_nonneg_left hlength (by positivity))
(by positivity) (by norm_num)
_ = 4*R*R^(-(T+3)) := by ring
_ ≤ R^3*R^(-(T+3)) := by
apply mul_le_mul_of_nonneg_right _ (by positivity)
nlinarith [sq_nonneg (R-2)]
_ = R^(-T) := by
rw [← Real.rpow_natCast R 3,← Real.rpow_add hR]
congr 1
norm_num
end PlanarFalconer
end OAI
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