openAI-math-dataset / lean /OAI /MeasureTheory /Falconer /Angular /AllDimensionalAngularProfileSum.lean
| import OAI.MeasureTheory.Falconer.Graphs.GraphNormalization | |
| import OAI.MeasureTheory.Falconer.Estimates.InversePowerSum | |
| namespace OAI | |
| /-! | |
| # Critical-dimensional cancellation in the parallel-test average | |
| Here `s` will be d/2. The interval minimum is over the original integer | |
| depths, and the retained profile is evaluated at those exact depths. | |
| -/ | |
| namespace Falconer | |
| theorem dyadic_power_rpow_as_R {N : ℕ} (hN : 0 < N) (n : ℕ) (s : ℝ) : | |
| ((2:ℝ)^n)^s = ((2:ℝ)^N)^(s*(n:ℝ)/N) := by | |
| rw [dyadic_rpow_change_base hN,← Real.rpow_natCast_mul (by norm_num : (0:ℝ) ≤ 2)] | |
| congr 1 | |
| ring | |
| theorem angular_profile_power_weight (m : ℝ → ℝ) {N : ℕ} (hN : 0 < N) | |
| (s E : ℝ) {g h p : ℕ} (hgh : g ≤ h) (hhp : h ≤ p) : | |
| (((2:ℝ)^N)^(m p-m h+E)*((2:ℝ)^h)^s)/((2:ℝ)^p)^s ≤ | |
| ((2:ℝ)^N)^(PlanarFalconer.height (fun n : ℕ => m n-s*(n:ℝ)/N) g p+E) := by | |
| have hp : 0 < (2:ℝ)^N := by positivity | |
| rw [dyadic_power_rpow_as_R hN h,dyadic_power_rpow_as_R hN p, | |
| ← Real.rpow_add hp,← Real.rpow_sub hp] | |
| apply Real.rpow_le_rpow_of_exponent_le (one_le_pow₀ (by norm_num)) | |
| have hh := PlanarFalconer.intervalMin_le (fun n : ℕ => m n-s*(n:ℝ)/N) hgh hhp | |
| dsimp only [PlanarFalconer.height] | |
| linarith | |
| theorem angular_profile_power_sum (m : ℝ → ℝ) {N : ℕ} (hN : 0 < N) | |
| (s E : ℝ) {g p : ℕ} (hgp : g ≤ p) : | |
| ((((2:ℝ)^N)^(m p-m g+E)*((2:ℝ)^g)^s + | |
| ∑ j ∈ Finset.range (p-g), | |
| ((2:ℝ)^N)^(m p-m (g+j)+E)*((2:ℝ)^(g+j+1))^s))/((2:ℝ)^p)^s ≤ | |
| (1+(2:ℝ)^s*(p-g:ℕ))* | |
| ((2:ℝ)^N)^(PlanarFalconer.height (fun n : ℕ => m n-s*(n:ℝ)/N) g p+E) := by | |
| let Z := ((2:ℝ)^N)^(PlanarFalconer.height (fun n : ℕ => m n-s*(n:ℝ)/N) g p+E) | |
| rw [add_div,Finset.sum_div] | |
| have hbase := angular_profile_power_weight m hN s E (g := g) le_rfl hgp | |
| have hsum : (∑ j ∈ Finset.range (p-g), | |
| (((2:ℝ)^N)^(m p-m (g+j)+E)*((2:ℝ)^(g+j+1))^s)/((2:ℝ)^p)^s) ≤ | |
| (p-g:ℕ)*((2:ℝ)^s*Z) := by | |
| calc | |
| _ ≤ ∑ _j ∈ Finset.range (p-g), ((2:ℝ)^s*Z) := by | |
| apply Finset.sum_le_sum | |
| intro j hj | |
| have hj' := Finset.mem_range.mp hj | |
| have hh := angular_profile_power_weight m hN s E (g := g) (h := g+j) (p := p) | |
| (by omega) (by omega) | |
| simp only [Nat.cast_add] at hh | |
| rw [pow_succ,Real.mul_rpow (by positivity) (by positivity)] | |
| have he : (((2:ℝ)^N)^(m p-m (g+j)+E)*(((2:ℝ)^(g+j))^s*(2:ℝ)^s))/((2:ℝ)^p)^s = | |
| (2:ℝ)^s*((((2:ℝ)^N)^(m p-m (g+j)+E)*((2:ℝ)^(g+j))^s)/((2:ℝ)^p)^s) := by ring | |
| rw [he] | |
| exact mul_le_mul_of_nonneg_left hh (by positivity) | |
| _ = _ := by simp | |
| change _ ≤ (1+(2:ℝ)^s*(p-g:ℕ))*Z | |
| change _ ≤ Z at hbase | |
| nlinarith | |
| end Falconer | |
| end OAI | |