| import OAI.MeasureTheory.Falconer.Shells.HighShellRadialBounds |
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| namespace OAI |
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| namespace PlanarFalconer |
| open Set MeasureTheory |
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| 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 |
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| end PlanarFalconer |
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| end OAI |
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