Project-declaredLean 4.33.0-rc1 · mathlib@79d0395a1825

Multi Dist of cast

multiDist_of_cast

additive combinatorics entropy probability bounding mutual

Plain-language statement

Multidistance is unchanged when a finite family of random variables is reindexed along an equality $m'=m$ . This says that the quantity depends on the family, not on the particular equal presentation of its finite index type.

Exact Lean statement

lemma multiDist_of_cast {m m' : ℕ} (h : m' = m) {Ω : Fin m → Type*}
    (hΩ : ∀ i, MeasureSpace (Ω i)) (hΩfin : ∀ i, IsFiniteMeasure (hΩ i).volume)
    {G : Type*} [MeasurableFinGroup G] (X : ∀ i, Ω i → G) :
    D[fun i ↦ X (i.cast h); fun i ↦ hΩ (i.cast h)] = D[X ; hΩ]

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

lemma multiDist_of_cast {m m' : ℕ} (h : m' = m) {Ω : Fin m → Type*}    (hΩ : ∀ i, MeasureSpace (Ω i)) (hΩfin : ∀ i, IsFiniteMeasure (hΩ i).volume)    {G : Type*} [MeasurableFinGroup G] (X : ∀ i, Ω i → G) :    D[fun i ↦ X (i.cast h); fun i ↦ hΩ (i.cast h)] = D[X ; hΩ] := by    unfold multiDist    congr 1    · apply IdentDistrib.entropy_congr      refine {        aemeasurable_fst := by fun_prop        aemeasurable_snd := by fun_prop        map_eq := ?_       }      have : (fun (x: Fin m' → G) ↦ ∑ i, x i) =          (fun (x: Fin m → G) ↦ ∑ i, x i) ∘ (fun (x: Fin m' → G) ↦ x ∘ (Fin.cast h.symm)) := by        ext x; dsimp; symm; apply Function.Bijective.sum_comp (Fin.cast_bijective h.symm)      rw [this, ← Measure.map_map] <;> try fun_prop      congr      exact Measure.pi_map_piCongrLeft (finCongr h) (fun i ↦ Measure.map (X i) ℙ)    congr 1    · rw [h]    convert Finset.sum_bijective _ (Fin.cast_bijective h) ?_ ?_ using 1 <;> simp

Project: Polynomial Freiman-Ruzsa project
License: Apache-2.0
Commit: a177b2e4abe4
Source: PFR/BoundingMutual.lean:16-36

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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