Project-declaredLean 4.33.0-rc1 · mathlib@79d0395a1825

Exists is Uniform of rdist self eq zero

exists_isUniform_of_rdist_self_eq_zero

additive combinatorics entropy probability hundred percent

Plain-language statement

If $d[X ;X]=0$ , then there exists a subgroup $H \leq G$ such that $d[X ;U_H] = 0$ .

Exact Lean statement

theorem exists_isUniform_of_rdist_self_eq_zero (hX : Measurable X) (hdist : d[X # X] = 0) :
    ∃ H : AddSubgroup G, ∃ U : Ω → G, Measurable U ∧ IsUniform H U ∧ d[X # U] = 0

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

theorem exists_isUniform_of_rdist_self_eq_zero (hX : Measurable X) (hdist : d[X # X] = 0) :    ∃ H : AddSubgroup G, ∃ U : Ω → G, Measurable U ∧ IsUniform H U ∧ d[X # U] = 0 := by  -- use for `U` a translate of `X` to make sure that `0` is in its support.  obtain ⟨x₀, h₀⟩ : ∃ x₀, ℙ (X ⁻¹' {x₀}) ≠ 0 := by    by_contra! h    have A a : (ℙ : Measure Ω).map X {a} = 0 := by      rw [Measure.map_apply hX .of_discrete]      exact h _    have B : (ℙ : Measure Ω).map X = 0 := by      rw [← Measure.sum_smul_dirac (μ := (ℙ : Measure Ω).map X)]      simp [A]    have : IsProbabilityMeasure ((ℙ : Measure Ω).map X) :=      Measure.isProbabilityMeasure_map hX.aemeasurable    exact IsProbabilityMeasure.ne_zero _ B  refine ⟨symmGroup X hX, fun ω ↦ X ω - x₀, hX.sub_const _,    isUniform_sub_const_of_rdist_eq_zero hX hdist h₀, ?_⟩  simp_rw [sub_eq_add_neg, ← Pi.add_def, rdist_add_const hX hX, hdist]

Project: Polynomial Freiman-Ruzsa project
License: Apache-2.0
Commit: a177b2e4abe4
Source: PFR/HundredPercent.lean:142-158

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Project documentation

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Person-level attribution pending.

View proof record

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Plain-language statement

If $A \subset {\bf F}_2^n$ is finite non-empty with $|A+A| \leq K|A|$ , then there exists a subgroup $H$ of ${\bf F}_2^n$ with $|H| \leq |A|$ such that $A$ can be covered by at most $2K^9$ translates of $H$ .

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Source project: Polynomial Freiman-Ruzsa project

Person-level attribution pending.

View proof record

Project-declaredLean 4.33.0-rc1

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Project documentation

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Source project: Polynomial Freiman-Ruzsa project

Person-level attribution pending.

View proof record