Project-declaredLean 4.33.0-rc1 · mathlib@79d0395a1825

Entropic PFR conjecture

entropic_PFR_conjecture

additive combinatorics entropy probability entropy pfr

Plain-language statement

entropic_PFR_conjecture: For two $G$ -valued random variables $X^0_1, X^0_2$ , there is some subgroup $H \leq G$ such that $d[X^0_1;U_H] + d[X^0_2;U_H] \le 11 d[X^0_1;X^0_2]$ .

Exact Lean statement

theorem entropic_PFR_conjecture (hpη : p.η = 1 / 9) :
    ∃ H : Submodule (ZMod 2) G, ∃ Ω : Type uG, ∃ mΩ : MeasureSpace Ω, ∃ U : Ω → G,
    IsProbabilityMeasure (ℙ : Measure Ω) ∧ Measurable U ∧
    IsUniform H U ∧ d[p.X₀₁ # U] + d[p.X₀₂ # U] ≤ 11 * d[p.X₀₁ # p.X₀₂]

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

theorem entropic_PFR_conjecture (hpη : p.η = 1 / 9) :    ∃ H : Submodule (ZMod 2) G, ∃ Ω : Type uG, ∃ mΩ : MeasureSpace Ω, ∃ U : Ω → G,    IsProbabilityMeasure (ℙ : Measure Ω) ∧ Measurable U ∧    IsUniform H U ∧ d[p.X₀₁ # U] + d[p.X₀₂ # U] ≤ 11 * d[p.X₀₁ # p.X₀₂] := by  cases nonempty_fintype G  obtain ⟨Ω', mΩ', X₁, X₂, hX₁, hX₂, _, htau_min⟩ := tau_minimizer_exists p  have hdist : d[X₁ # X₂] = 0 := tau_strictly_decreases p hX₁ hX₂ htau_min hpη  obtain ⟨H, U, hU, hH_unif, hdistX₁, hdistX₂⟩ := exists_isUniform_of_rdist_eq_zero hX₁ hX₂ hdist  refine ⟨AddSubgroup.toZModSubmodule _ H, Ω', inferInstance, U, inferInstance, hU, hH_unif , ?_⟩  have h : τ[X₁ # X₂ | p] ≤ τ[p.X₀₂ # p.X₀₁ | p] := is_tau_min p htau_min p.hmeas2 p.hmeas1  rw [tau, tau, hpη] at h  norm_num at h  have : d[p.X₀₁ # p.X₀₂] = d[p.X₀₂ # p.X₀₁] := rdist_symm  have : d[p.X₀₁ # U] ≤ d[p.X₀₁ # X₁] + d[X₁ # U] := rdist_triangle p.hmeas1 hX₁ hU  have : d[p.X₀₂ # U] ≤ d[p.X₀₂ # X₂] + d[X₂ # U] := rdist_triangle p.hmeas2 hX₂ hU  linarith

Project: Polynomial Freiman-Ruzsa project
License: Apache-2.0
Commit: a177b2e4abe4
Source: PFR/EntropyPFR.lean:52-67

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Related declarations

Project-declaredLean 4.33.0-rc1

Approx hom pfr

approx_hom_pfr

Project documentation

An approximate-homomorphism theorem for finite elementary abelian $2$ -groups. Let $f:G\to G'$ and $K>0$ . If at least a proportion $K^{-1}$ of pairs $(x,y)\in G^2$ satisfy $f(x+y)=f(x)+f(y)$ , then there are an additive homomorphism $\varphi:G\to G'$ and a constant $c\in G'$ such that $f(x)=\varphi(x)+c$ for at least $|G|/(2^{144}K^{122})$ values of $x$ .

additive combinatoricsentropyprobability

Source project: Polynomial Freiman-Ruzsa project

Person-level attribution pending.

View proof record

Project-declaredLean 4.33.0-rc1

Better PFR conjecture

better_PFR_conjecture

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 $H$ .

additive combinatoricsentropyprobability

Source project: Polynomial Freiman-Ruzsa project

Person-level attribution pending.

View proof record

Project-declaredLean 4.33.0-rc1

Better PFR conjecture

better_PFR_conjecture'

Project documentation

Polynomial Freiman-Ruzsa theorem with exponent $9$ , without a finite ambient-group assumption. Let $A$ be a nonempty finite subset of an elementary abelian $2$ -group. If $|A+A|\le K|A|$ , then there are a finite subspace $H$ and a finite set $c$ such that $A\subseteq c+H$ , $|c|<2K^9$ , and $|H|\le|A|$ .

additive combinatoricsentropyprobability

Source project: Polynomial Freiman-Ruzsa project

Person-level attribution pending.

View proof record