| import OAI.NumberTheory.PiExponent.Analysis.AnalyticCollision |
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| namespace OAI |
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| open scoped BigOperators |
| open Complex Finset MeasureTheory Filter |
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| namespace PiExponent.AnalyticCollision |
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| 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 _ |
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| 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 |
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| 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 _ |
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| end PiExponent.AnalyticCollision |
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| end OAI |
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