Research proof index

Let $f: G \to G'$ be a function, and let $S$ denote the set $S := \{ f(x+y)-f(x)-f(y): x,y \in G \}.$ Then there exists a homomorphism $\phi: G \to G'$ such that $|\{f(x) - \phi(x)\}| \leq |S|^{10}.$

A symmetry identity in the $\tau$-minimizer endgame. Let $X_1',X_2'$ be independent copies of $X_1,X_2$, and set $U=X_1+X_2$, $V=X_1'+X_2$, $W=X_1'+X_1$, and $S=X_1+X_2+X_1'+X_2'$. Then the conditional mutual informations agree: $I[V:W\mid S]=I[U:W\mid S]$.

If $X, Y, Z$ are independent $G$-valued random variables, then $D_{KL}(X+Z\Vert Y+Z) \leq D_{KL}(X\Vert Y).$

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.

Let m ≥ 2, and let X_[m] be a tuple of G-valued random variables. Let W := ∑ X_i. Then d[W;-W] ≤ 2 D[X_i].

If $(X_i)_{1 \leq i \leq m}$ is a $\tau$-minimizer, then $\sum_{i=1}^m d[X_i; X^0] \leq \frac{2m}{\eta} d[X^0; X^0]$.