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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 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 | import OAI.NumberTheory.PiExponent.Analysis.LogTailAnalytic
import OAI.NumberTheory.PiExponent.Approximation.MatrixTranslation
namespace OAI
open scoped BigOperators
open PiExponent.AnalyticCollision PiExponent.LogTailAnalytic
namespace PiExponent.PeriodAnalytic
noncomputable def periodSeries (b center : ℂ) (h d : ℕ) : PowerSeries ℂ :=
PowerSeries.C b * (1 + PowerSeries.X) ^ h *
(PowerSeries.C center + PowerSeries.log ℂ) ^ d
noncomputable def periodFunction (b center : ℂ) (h d : ℕ) (t : ℂ) : ℂ :=
b * (1 + t) ^ h * (center + Complex.log (1 + t)) ^ d
theorem hasSum_coeff_C (b z : ℂ) :
HasSum (fun n => PowerSeries.coeff n (PowerSeries.C b) * z ^ n) b := by
convert hasSum_ite_eq 0 b using 1
funext n
by_cases hn : n = 0 <;> simp [PowerSeries.coeff_C, hn]
theorem hasSum_coeff_X (z : ℂ) :
HasSum (fun n => PowerSeries.coeff n (PowerSeries.X : PowerSeries ℂ) * z ^ n) z := by
convert hasSum_ite_eq 1 z using 1
funext n
by_cases hn : n = 1 <;> simp [PowerSeries.coeff_X, hn]
theorem hasSum_coeff_add {f g : PowerSeries ℂ} {x y z : ℂ}
(hf : HasSum (fun n => PowerSeries.coeff n f * z ^ n) x)
(hg : HasSum (fun n => PowerSeries.coeff n g * z ^ n) y) :
HasSum (fun n => PowerSeries.coeff n (f + g) * z ^ n) (x + y) := by
simpa only [map_add, add_mul] using hf.add hg
theorem hasSum_periodSeries (b center : ℂ) (h d : ℕ) {z : ℂ} (hz : ‖z‖ < 1) :
HasSum (fun n => PowerSeries.coeff n (periodSeries b center h d) * z ^ n)
(periodFunction b center h d z) := by
exact hasSum_coeff_mul
(hasSum_coeff_mul (hasSum_coeff_C b z)
(hasSum_coeff_pow (hasSum_coeff_add (hasSum_coeff_one z) (hasSum_coeff_X z)) h))
(hasSum_coeff_pow (hasSum_coeff_add (hasSum_coeff_C center z) (hasSum_formal_log hz)) d)
theorem analyticAt_periodFunction (b center : ℂ) (h d : ℕ) {z : ℂ} (hz : ‖z‖ < 1) :
AnalyticAt ℂ (periodFunction b center h d) z := by
unfold periodFunction
apply AnalyticAt.mul
· fun_prop
· exact (analyticAt_const.add ((analyticAt_const.add analyticAt_id).clog
(Complex.mem_slitPlane_of_norm_lt_one hz))).pow d
theorem hasFPowerSeriesAt_periodFunction (b center : ℂ) (h d : ℕ) :
HasFPowerSeriesAt (periodFunction b center h d)
(FormalMultilinearSeries.ofScalars ℂ (fun n => PowerSeries.coeff n
(periodSeries b center h d))) 0 := by
rw [hasFPowerSeriesAt_iff]
filter_upwards [Metric.ball_mem_nhds (0 : ℂ) (by norm_num : (0 : ℝ) < 1)] with z hz
have hn : ‖z‖ < 1 := by simpa only [Metric.mem_ball, dist_zero_right] using hz
simpa only [FormalMultilinearSeries.coeff_ofScalars, zero_add, smul_eq_mul, mul_comm] using
hasSum_periodSeries b center h d hn
theorem rowTest_periodFunction_eq_coeff (b center : ℂ) (h d ell : ℕ) :
rowTest ell (periodFunction b center h d) =
PowerSeries.coeff ell (periodSeries b center h d) := by
have hd : DifferentiableOn ℂ (periodFunction b center h d)
(Metric.closedBall 0 (1 / 2)) := by
intro z hz
apply (analyticAt_periodFunction b center h d ?_).differentiableAt.differentiableWithinAt
have hz' : ‖z‖ ≤ 1 / 2 := by simpa only [Metric.mem_closedBall, dist_zero_right] using hz
linarith
have hp := hd.hasFPowerSeriesOnBall (R := (1 / 2 : NNReal)) (by norm_num)
have he := hp.hasFPowerSeriesAt.eq_formalMultilinearSeries
(hasFPowerSeriesAt_periodFunction b center h d)
have hh := congrArg (fun p : FormalMultilinearSeries ℂ ℂ ℂ => p ell (fun _ => 1)) he
simpa only [rowTest, FormalMultilinearSeries.apply_eq_prod_smul_coeff,
Finset.prod_const_one, one_smul, FormalMultilinearSeries.coeff_ofScalars,
NNReal.coe_div, NNReal.coe_one, NNReal.coe_ofNat] using hh
theorem rowTest_congr_sphere {f g : ℂ → ℂ}
(hfg : ∀ z ∈ Metric.sphere (0 : ℂ) (1 / 2), f z = g z) (ell : ℕ) :
rowTest ell f = rowTest ell g := by
simp only [rowTest, cauchyPowerSeries_apply]
congr 1
apply circleIntegral.integral_congr (by norm_num)
intro z hz
change (1 / (z - 0)) ^ ell • (z - 0)⁻¹ • f z =
(1 / (z - 0)) ^ ell • (z - 0)⁻¹ • g z
rw [hfg z hz]
theorem periodFunction_eq_exponentialMonomial (b center : ℂ) (h d : ℕ)
(hcenter : Complex.exp center = 1) {z : ℂ} (hz : ‖z‖ < 1) :
periodFunction b center h d z =
exponentialMonomial b (h : ℂ) d (center + Complex.log (1 + z)) := by
have hz0 : 1 + z ≠ 0 := by
intro heq
have heq' : z = -1 := by linear_combination heq
simp [heq'] at hz
unfold periodFunction exponentialMonomial
rw [Complex.exp_nat_mul, Complex.exp_add, hcenter, Complex.exp_log hz0, one_mul]
theorem exp_period_center (j : ℕ) :
Complex.exp ((j : ℂ) * (2 * (Real.pi : ℂ) * Complex.I)) = 1 := by
rw [Complex.exp_nat_mul, Complex.exp_two_pi_mul_I, one_pow]
theorem rowTest_exponentialMonomial_eq_coeff (b : ℂ) (j h d ell : ℕ) :
rowTest ell (fun t => exponentialMonomial b (h : ℂ) d
((j : ℂ) * (2 * (Real.pi : ℂ) * Complex.I) + Complex.log (1 + t))) =
PowerSeries.coeff ell (periodSeries b ((j : ℂ) * (2 * (Real.pi : ℂ) * Complex.I)) h d) := by
rw [← rowTest_periodFunction_eq_coeff]
apply rowTest_congr_sphere
intro t ht
symm
apply periodFunction_eq_exponentialMonomial _ _ _ _ (exp_period_center j)
have hn : ‖t‖ = (1 / 2 : ℝ) := by simpa only [Metric.mem_sphere, dist_zero_right] using ht
rw [hn]
norm_num
theorem periodMonomial_coeff_eq_rowTest {m : ℕ} (j h ell : ℕ)
(a : Fin m → ℕ) (A : Fin m →₀ ℕ) :
PowerSeries.coeff ell ((MatrixTranslation.periodMonomial j
(2 * (Real.pi : ℂ) * Complex.I) h a).coeff A) =
rowTest ell (fun t => exponentialMonomial
((∏ i, (a i).choose (A i) : ℕ) : ℂ) (h : ℂ) (∑ i : Fin m, (a i - A i))
((j : ℂ) * (2 * (Real.pi : ℂ) * Complex.I) + Complex.log (1 + t))) := by
rw [rowTest_exponentialMonomial_eq_coeff, MatrixTranslation.periodMonomial_coeff]
simp only [periodSeries, MatrixTranslation.periodCoordinate, map_natCast]
theorem norm_binomial_product_le {m : ℕ} (a : Fin m → ℕ) (A : Fin m →₀ ℕ) :
‖((∏ i, (a i).choose (A i) : ℕ) : ℂ)‖ ≤ (2 : ℝ) ^ (∑ i, a i) := by
rw [Complex.norm_natCast]
have h : (∏ i, (a i).choose (A i) : ℕ) ≤ 2 ^ (∑ i, a i) := by
calc
_ ≤ ∏ i, 2 ^ a i := Finset.prod_le_prod (fun i _ => Nat.choose_le_two_pow _ _)
_ = _ := Finset.prod_pow_eq_pow_sum _ _ _
exact_mod_cast h
noncomputable def columnFunction {m : ℕ} (h : ℕ) (a : Fin m → ℕ)
(A : Fin m →₀ ℕ) : ℂ → ℂ :=
exponentialMonomial ((∏ i, (a i).choose (A i) : ℕ) : ℂ) (h : ℂ)
(∑ i : Fin m, (a i - A i))
theorem differentiable_columnFunction {m : ℕ} (h : ℕ) (a : Fin m → ℕ)
(A : Fin m →₀ ℕ) : Differentiable ℂ (columnFunction h a A) :=
differentiable_exponentialMonomial _ _ _
theorem norm_columnFunction_le {m : ℕ} (h : ℕ) (a : Fin m → ℕ)
(A : Fin m →₀ ℕ) (w : Fin m → ℝ) {R H w0 wstar : ℝ}
(hR : 1 ≤ R) (hH : 0 ≤ H) (hw0 : 0 < w0) (hws : 0 < wstar)
(hw : ∀ i, wstar ≤ w i) (hcol : w0 * h + ∑ i, w i * a i ≤ H)
{z : ℂ} (hz : ‖z‖ ≤ R) :
‖columnFunction h a A z‖ ≤
Real.exp (H * (R / w0 + Real.log (2 * R) / wstar)) := by
have hwpos (i) : 0 ≤ w i := hws.le.trans (hw i)
have hsum0 : 0 ≤ ∑ i, w i * a i := Finset.sum_nonneg fun i _ => mul_nonneg (hwpos i) (Nat.cast_nonneg _)
have hh : ‖(h : ℂ)‖ ≤ H / w0 := by
rw [Complex.norm_natCast, le_div_iff₀ hw0]
nlinarith
have htot : (((∑ i, a i : ℕ) : ℝ)) ≤ H / wstar := by
rw [le_div_iff₀ hws]
have hs : wstar * ∑ i, (a i : ℝ) ≤ ∑ i, w i * a i := by
rw [Finset.mul_sum]
exact Finset.sum_le_sum fun i _ => mul_le_mul_of_nonneg_right (hw i) (Nat.cast_nonneg _)
have hcast : ((∑ i, a i : ℕ) : ℝ) = ∑ i, (a i : ℝ) := by push_cast; rfl
rw [hcast]
nlinarith
exact norm_exponentialMonomial_le_exp_budget _ _ _ _ hR hH hw0 hws hz
(norm_binomial_product_le a A) hh
(Finset.sum_le_sum fun i _ => Nat.sub_le _ _) htot
end PiExponent.PeriodAnalytic
end OAI
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