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cdec422 | 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 | import Mathlib.Analysis.Complex.Basic
import Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots
namespace OAI
/-!
# Finite root-of-unity filters
Averaging the integer powers of a primitive root over one period detects
exactly the exponents divisible by that period. This is the finite Fourier
filter used for auxiliary separation.
-/
open scoped BigOperators
namespace MatrixMultiplication.AuxiliarySeparation
/-- The sum over a complete period of a root of unity is zero unless the root is one. -/
lemma sum_pow_root_of_unity (z : ℂ) (L : ℕ) (hz : z ^ L = 1) :
(∑ r ∈ Finset.range L, z ^ r) = if z = 1 then (L : ℂ) else 0 := by
split_ifs with h
· simp [h]
· have hsum : (1 - z) * ∑ r ∈ Finset.range L, z ^ r = 0 := by
rw [mul_neg_geom_sum, hz, sub_self]
exact (mul_eq_zero.mp hsum).resolve_left (sub_ne_zero.mpr (Ne.symm h))
/-- An unnormalized finite Fourier filter, with an integer exponent. -/
theorem sum_zpow_primitive_root {ζ : ℂ} {L : ℕ}
(hζ : IsPrimitiveRoot ζ L) (e : ℤ) :
(∑ r ∈ Finset.range L, ζ ^ ((r : ℤ) * e)) =
if (L : ℤ) ∣ e then (L : ℂ) else 0 := by
have hpow : (ζ ^ e) ^ L = 1 := by
rw [← zpow_natCast, ← zpow_mul, mul_comm, zpow_mul, hζ.zpow_eq_one, one_zpow]
have hterm (r : ℕ) : ζ ^ ((r : ℤ) * e) = (ζ ^ e) ^ r := by
rw [← zpow_natCast, ← zpow_mul, mul_comm]
simp_rw [hterm]
rw [sum_pow_root_of_unity _ _ hpow]
simp only [hζ.zpow_eq_one_iff_dvd]
/-- Averaging over a primitive root's period selects exactly its divisible exponents. -/
theorem normalized_sum_zpow_primitive_root {ζ : ℂ} {L : ℕ}
(hL : L ≠ 0) (hζ : IsPrimitiveRoot ζ L) (e : ℤ) :
(L : ℂ)⁻¹ * (∑ r ∈ Finset.range L, ζ ^ ((r : ℤ) * e)) =
if (L : ℤ) ∣ e then 1 else 0 := by
rw [sum_zpow_primitive_root hζ e]
split_ifs <;> simp [hL]
/-- Divisibility by a positive period is exact equality to zero inside one period. -/
lemma int_dvd_iff_eq_zero_of_abs_lt {L : ℕ} {e : ℤ} (he : |e| < (L : ℤ)) :
(L : ℤ) ∣ e ↔ e = 0 := by
constructor
· intro hd
apply abs_eq_zero.mp
exact Int.eq_zero_of_dvd_of_nonneg_of_lt (abs_nonneg e) he
((dvd_abs (L : ℤ) e).mpr hd)
· rintro rfl
exact dvd_zero _
/-- Within one period, the finite Fourier filter selects exactly exponent zero. -/
theorem normalized_sum_zpow_primitive_root_of_abs_lt {ζ : ℂ} {L : ℕ} {e : ℤ}
(hL : L ≠ 0) (hζ : IsPrimitiveRoot ζ L) (he : |e| < (L : ℤ)) :
(L : ℂ)⁻¹ * (∑ r ∈ Finset.range L, ζ ^ ((r : ℤ) * e)) =
if e = 0 then 1 else 0 := by
rw [normalized_sum_zpow_primitive_root hL hζ e]
simp only [int_dvd_iff_eq_zero_of_abs_lt he]
end MatrixMultiplication.AuxiliarySeparation
end OAI
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