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a72760f | 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 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 | import OAI.NumberTheory.PiExponent.Analysis.AnalyticCollision
namespace OAI
open scoped BigOperators
open Complex Finset MeasureTheory Filter
namespace PiExponent.AnalyticCollision
theorem hasSum_rowTest {F : ℕ → ℂ → ℂ} {f : ℂ → ℂ} {bound : ℕ → ℝ}
(hF : ∀ n, Measurable (F n))
(hbound : ∀ n z, z ∈ Metric.sphere (0 : ℂ) (1 / 2) → ‖F n z‖ ≤ bound n)
(hsum : Summable bound)
(hpoint : ∀ z ∈ Metric.sphere (0 : ℂ) (1 / 2), HasSum (fun n => F n z) (f z))
(ell : ℕ) : HasSum (fun n => rowTest ell (F n)) (rowTest ell f) := by
simp only [rowTest, cauchyPowerSeries_apply]
apply HasSum.const_smul
change HasSum (fun n => ∫ θ in (0 : ℝ)..2 * Real.pi,
deriv (circleMap 0 (1 / 2)) θ •
((1 / (circleMap 0 (1 / 2) θ - 0)) ^ ell •
(circleMap 0 (1 / 2) θ - 0)⁻¹ • F n (circleMap 0 (1 / 2) θ)))
(∫ θ in (0 : ℝ)..2 * Real.pi,
deriv (circleMap 0 (1 / 2)) θ •
((1 / (circleMap 0 (1 / 2) θ - 0)) ^ ell •
(circleMap 0 (1 / 2) θ - 0)⁻¹ • f (circleMap 0 (1 / 2) θ)))
apply intervalIntegral.hasSum_integral_of_dominated_convergence
(fun n _ => (2 : ℝ) ^ ell * bound n)
· intro n
apply Measurable.aestronglyMeasurable
simp only [deriv_circleMap]
exact (by fun_prop)
· intro n
apply Eventually.of_forall
intro θ _
have hb := hbound n (circleMap 0 (1 / 2) θ) (circleMap_mem_sphere 0 (by norm_num) θ)
simp only [deriv_circleMap, norm_smul, norm_mul, norm_I, mul_one, sub_zero,
norm_pow, norm_div, norm_one, norm_inv, norm_circleMap_zero]
norm_num only [abs_of_pos (by norm_num : (0 : ℝ) < 1 / 2), one_div, inv_div, div_one]
calc
_ = (2 : ℝ) ^ ell * ‖F n (circleMap 0 (1 / 2) θ)‖ := by ring
_ ≤ (2 : ℝ) ^ ell * bound n := mul_le_mul_of_nonneg_left hb (by positivity)
· exact Eventually.of_forall fun _ _ => hsum.mul_left _
· simpa only [tsum_mul_left] using
(intervalIntegrable_const : IntervalIntegrable (fun _ : ℝ => (2 : ℝ) ^ ell * ∑' n, bound n)
volume 0 (2 * Real.pi))
· apply Eventually.of_forall
intro θ _
exact (((hpoint _ (circleMap_mem_sphere 0 (by norm_num) θ)).const_smul _).const_smul _).const_smul _
theorem rowTest_const_mul (ell : ℕ) (c : ℂ) (f : ℂ → ℂ) :
rowTest ell (fun t => c * f t) = c * rowTest ell f := by
simp only [rowTest, cauchyPowerSeries_apply, smul_eq_mul]
have heq : (fun z : ℂ => (1 / (z - 0)) ^ ell * ((z - 0)⁻¹ * (c * f z))) =
(fun z : ℂ => c * ((1 / (z - 0)) ^ ell * ((z - 0)⁻¹ * f z))) := by
funext z
ring
rw [heq, circleIntegral.integral_const_mul]
ring
theorem hasSum_translated_row {f : ℂ → ℂ} (hf : Differentiable ℂ f)
{R : NNReal} (hR : 0 < R) {D : ℝ} (hD : 0 ≤ D)
(hfD : ∀ z ∈ Metric.sphere (0 : ℂ) (R : ℝ), ‖f z‖ ≤ D)
{center : ℂ} (hc : ‖center‖ + 3 / 4 ≤ (R : ℝ) / 2) (ell : ℕ) :
HasSum (fun d => rowTest ell (fun t => ((center + Complex.log (1 + t)) / (R : ℂ)) ^ d) *
normalizedTaylorCoeff f R d)
(rowTest ell (fun t => f (center + Complex.log (1 + t)))) := by
have h := hasSum_rowTest
(F := fun d t => normalizedTaylorCoeff f R d *
((center + Complex.log (1 + t)) / (R : ℂ)) ^ d)
(f := fun t => f (center + Complex.log (1 + t)))
(bound := fun d => D * (1 / 2 : ℝ) ^ d)
(by intro d; fun_prop) ?_ ?_ ?_ ell
· convert! h using 1
funext d
rw [rowTest_const_mul, mul_comm]
· intro d t ht
rw [norm_mul, norm_pow]
apply mul_le_mul (norm_normalizedTaylorCoeff_le hR hfD d)
· apply pow_le_pow_left₀ (norm_nonneg _) (norm_shifted_log_div_le_half hR hc _)
simpa only [Metric.mem_sphere, dist_zero_right] using ht.le
· positivity
· exact hD
· exact (summable_geometric_of_lt_one (by norm_num : (0 : ℝ) ≤ 1 / 2)
(by norm_num : (1 / 2 : ℝ) < 1)).mul_left D
· intro t _
exact hasSum_normalizedTaylorCoeff hf hR _
end PiExponent.AnalyticCollision
end OAI
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