Project-declaredLean 4.32.0 · mathlib@81a5d257c8e4

Le iff width

BohrSet.le_iff_width

additive combinatorics arithmetic progressions Fourier analysis basic

Plain-language statement

Characterization of the order on Bohr sets. The relation $B_1\le B_2$ holds exactly when every frequency of $B_2$ is also a frequency of $B_1$ , and $\operatorname{width}_{B_1}(\psi)\le \operatorname{width}_{B_2}(\psi)$ for each such frequency.

Exact Lean statement

lemma le_iff_width {B₁ B₂ : BohrSet G} : B₁ ≤ B₂ ↔
    B₂.frequencies ⊆ B₁.frequencies ∧ ∀ ⦃ψ⦄, ψ ∈ B₂.frequencies → B₁.width ψ ≤ B₂.width ψ

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

lemma le_iff_width {B₁ B₂ : BohrSet G} : B₁ ≤ B₂ ↔    B₂.frequencies ⊆ B₁.frequencies ∧ ∀ ⦃ψ⦄, ψ ∈ B₂.frequencies → B₁.width ψ ≤ B₂.width ψ := by  constructor  case mp =>    intro h    refine ⟨frequencies_anti h, fun ψ hψ => ?_⟩    rw [←ENNReal.coe_le_coe, coe_width hψ, coe_width (frequencies_anti h hψ)]    exact h ψ  case mpr =>    rintro ⟨h₁, h₂⟩ ψ    by_cases ψ ∈ B₂.frequencies    case neg h' => simp [ewidth_eq_top_of_not_mem_frequencies h']    case pos h' =>      rw [←coe_width h', ←coe_width (h₁ h'), ENNReal.coe_le_coe]      exact h₂ h'

Project: Arithmetic Progressions Almost Periodicity
License: Apache-2.0
Commit: afafc42a5326
Source: APAP/Prereqs/Bohr/Basic.lean:168-182

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