Project-declaredLean 4.33.0-rc1 · mathlib@79d0395a1825

Cond Multi Dist of cast

condMultiDist_of_cast

additive combinatorics entropy probability bounding mutual

Plain-language statement

Conditional multidistance is unchanged when both the random variables and their conditioning variables are reindexed along an equality $m'=m$ . As with ordinary multidistance, the value does not depend on the chosen equal presentation of the finite index type.

Exact Lean statement

lemma condMultiDist_of_cast {m m' : ℕ} (h : m' = m) {Ω : Fin m → Type*}
    (hΩ : ∀ i, MeasureSpace (Ω i))
    {G S : Type*} [MeasurableFinGroup G] [Fintype S] (X : ∀ i, Ω i → G) (Y : ∀ i, Ω i → S) :
    D[fun i ↦ X (i.cast h) | fun i ↦ Y (i.cast h); fun i ↦ hΩ (i.cast h)] =
    D[X | Y ; hΩ]

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

lemma condMultiDist_of_cast {m m' : ℕ} (h : m' = m) {Ω : Fin m → Type*}    (hΩ : ∀ i, MeasureSpace (Ω i))    {G S : Type*} [MeasurableFinGroup G] [Fintype S] (X : ∀ i, Ω i → G) (Y : ∀ i, Ω i → S) :    D[fun i ↦ X (i.cast h) | fun i ↦ Y (i.cast h); fun i ↦ hΩ (i.cast h)] =    D[X | Y ; hΩ] := by  unfold condMultiDist  let ι : (Fin m' → S) → (Fin m → S) := fun x i ↦ x (i.cast h.symm)  have hι : Function.Bijective ι := by    constructor    · intro f g h'; ext i; replace h' := congrFun h' (i.cast h); simpa [ι] using h'    intro f; use f ∘ (Fin.cast h); ext i; simp [ι]  convert Function.Bijective.sum_comp hι _ with y _  congr 1  · convert Function.Bijective.prod_comp (Fin.cast_bijective h) _ with i _    rfl  convert multiDist_of_cast h _ _ X with i hmes i  · simp; congr  intros; simp; infer_instance

Project: Polynomial Freiman-Ruzsa project
License: Apache-2.0
Commit: a177b2e4abe4
Source: PFR/BoundingMutual.lean:77-94

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Project-declaredLean 4.33.0-rc1

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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

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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 $H$ .

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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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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