Project-declaredLean 4.32.0 · mathlib@81a5d257c8e4

Expect i Inf ker eq expect ite

expect_iInf_ker_eq_expect_ite

additive combinatorics arithmetic progressions Fourier analysis discrete

Plain-language statement

Let $V$ be the intersection of the kernels of a set of additive characters $\Delta$ . Averaging $f$ over $V$ equals the average of $\widehat f(\psi)$ over all characters, with the Fourier coefficient retained precisely when $\psi$ lies in the additive subgroup generated by $\Delta$ .

Exact Lean statement

lemma expect_iInf_ker_eq_expect_ite {Δ : Set (AddChar G ℂ)}
    [DecidablePred (· ∈ AddSubgroup.closure Δ)] {V : AddSubgroup G} [Fintype V]
    (hV : V = ⨅ γ ∈ Δ, γ.toAddMonoidHom.ker) (f : G → ℂ) :
    𝔼 x ∈ Set.toFinset V, f x = 𝔼 ψ, if ψ ∈ AddSubgroup.closure Δ then dft f ψ else 0

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

lemma expect_iInf_ker_eq_expect_ite {Δ : Set (AddChar G ℂ)}    [DecidablePred (· ∈ AddSubgroup.closure Δ)] {V : AddSubgroup G} [Fintype V]    (hV : V = ⨅ γ ∈ Δ, γ.toAddMonoidHom.ker) (f : G → ℂ) :    𝔼 x ∈ Set.toFinset V, f x = 𝔼 ψ, if ψ ∈ AddSubgroup.closure Δ then dft f ψ else 0 := by  rw [show (𝔼 x ∈ Set.toFinset V, f x) = 𝔼 x ∈ Set.toFinset V, 𝔼 ψ, dft f ψ * ψ x by    congr! with x    exact (dft_inversion f x).symm]  rw [expect_comm]  congr! with ψ  split_ifs with hψ  · rw [← mul_expect, AddChar.expect_iInf_ker_eq_one_of_mem_closure hV hψ, mul_one]  · rw [← mul_expect, AddChar.expect_iInf_ker_eq_zero_of_not_mem_closure hV hψ, mul_zero]

Project: Arithmetic Progressions Almost Periodicity
License: Apache-2.0
Commit: afafc42a5326
Source: APAP/Prereqs/FourierTransform/Discrete.lean:141-152

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