Project-declaredLean 4.33.0-rc1 · mathlib@79d0395a1825

Exists is Uniform of rdist eq zero

exists_isUniform_of_rdist_eq_zero

additive combinatorics entropy probability hundred percent

Plain-language statement

If $d[X_1;X_2]=0$ , then there exists a subgroup $H \leq G$ such that $d[X_1;U_H] = d[X_2;U_H] = 0$ . Follows from the preceding claim by the triangle inequality.

Exact Lean statement

theorem exists_isUniform_of_rdist_eq_zero
    {Ω' : Type*} [MeasureSpace Ω'] [IsProbabilityMeasure (ℙ : Measure Ω')] {X' : Ω' → G}
    (hX : Measurable X) (hX' : Measurable X') (hdist : d[X # X'] = 0) :
    ∃ H : AddSubgroup G, ∃ U : Ω → G,
      Measurable U ∧ IsUniform H U ∧ d[X # U] = 0 ∧ d[X' # U] = 0

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

theorem exists_isUniform_of_rdist_eq_zero    {Ω' : Type*} [MeasureSpace Ω'] [IsProbabilityMeasure (ℙ : Measure Ω')] {X' : Ω' → G}    (hX : Measurable X) (hX' : Measurable X') (hdist : d[X # X'] = 0) :    ∃ H : AddSubgroup G, ∃ U : Ω → G,      Measurable U ∧ IsUniform H U ∧ d[X # U] = 0 ∧ d[X' # U] = 0 := by  have h' : d[X # X] = 0 := by    apply le_antisymm _ (rdist_nonneg hX hX)    calc      d[X # X] ≤ d[X # X'] + d[X' # X] := rdist_triangle hX hX' hX      _ = 0 := by rw [hdist, rdist_symm, hdist, zero_add]  rcases exists_isUniform_of_rdist_self_eq_zero hX h' with ⟨H, U, hmeas, hunif, hd⟩  refine ⟨H, U, hmeas, hunif, hd, ?_⟩  apply le_antisymm _ (rdist_nonneg hX' hmeas)  calc    d[X' # U] ≤ d[X' # X] + d[X # U] := rdist_triangle hX' hX hmeas    _ = 0 := by rw [hd, rdist_symm, hdist, zero_add]

Project: Polynomial Freiman-Ruzsa project
License: Apache-2.0
Commit: a177b2e4abe4
Source: PFR/HundredPercent.lean:162-177

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Project documentation

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Person-level attribution pending.

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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.

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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