Research proof index

If G is torsion-free and X, Y are G-valued random variables then d[X; 2Y] ≤ 5d[X; Y].

Polynomial Freiman-Ruzsa theorem for bounded-torsion groups. Let $G$ be a finite abelian group in which $mx=0$ for every $x$, with $m\ge2$. If $A\subseteq G$ is nonempty and $|A+A|\le K|A|$, then there are a subgroup $H\le G$ and a set $c$ such that $A\subseteq c+H$, $|H|\le|A|$, and $|c|<mK^{256m^3+1}$.

The CWSS shape of the canonical ℓᵢ = 1 structure CWSSStructure.ofSpecialSound k is exactly the plain special-soundness shape distinctShape k. This is the structural heart of the equivalence between CWSS and plain special soundness: both the arity (1·(kᵢ-1)+1 = kᵢ) and the node predicate (IsSpecialSoundFamily 1 kᵢ vs. Function.Injective) agree.

For any open U ⊆ GL₂(𝔸_F), Dˣ\GL₂(𝔸_F)/U is finite. (where Dˣ is viewed as a subgroup of GL₂(𝔸_F) under the identification M₂(𝔸_F) ≃ D ⊗ 𝔸_F)

Hecke operators are preserved under the identification 𝒮²(U, χ; M) ≃ M ⊗ 𝒮²(U, χ; R).

Each coset in U1diagU1 is of the form unipotent_mul_diagU1 for some t ∈ O_v.