Project-declaredLean 4.33.0-rc1 · mathlib@79d0395a1825

Entropic PFR conjecture

entropic_PFR_conjecture'

additive combinatorics entropy probability entropy pfr

Plain-language statement

In the project's entropic PFR package with parameter $\eta=1/9$ , there is a subspace $H$ and a random variable $U$ uniformly distributed on $H$ such that each reference variable is within six times their mutual Ruzsa distance of $U$ : $d(X_1,U),d(X_2,U)\le6d(X_1,X_2)$ .

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,
    IsUniform H U ∧ d[p.X₀₁ # U] ≤ 6 * d[p.X₀₁ # p.X₀₂] ∧
      d[p.X₀₂ # U] ≤ 6 * 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,    IsUniform H U ∧ d[p.X₀₁ # U] ≤ 6 * d[p.X₀₁ # p.X₀₂] ∧      d[p.X₀₂ # U] ≤ 6 * d[p.X₀₁ # p.X₀₂] := by  have : d[p.X₀₁ # p.X₀₂] = d[p.X₀₂ # p.X₀₁] := rdist_symm  obtain ⟨H, Ω, mΩ, U, H', hU, hUnif, h'⟩ := entropic_PFR_conjecture p hpη  refine ⟨H, Ω, mΩ, U, hUnif, ?_⟩  have : d[p.X₀₁ # U] ≤ d[p.X₀₁ # p.X₀₂] + d[p.X₀₂ # U] := rdist_triangle p.hmeas1 p.hmeas2 hU  have : d[p.X₀₂ # U] ≤ d[p.X₀₂ # p.X₀₁] + d[p.X₀₁ # U] := rdist_triangle p.hmeas2 p.hmeas1 hU  constructor  · linarith  · linarith

Project: Polynomial Freiman-Ruzsa project
License: Apache-2.0
Commit: a177b2e4abe4
Source: PFR/EntropyPFR.lean:69-80

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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