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 $D_{KL}(X+Z\Vert Y+Z) \leq D_{KL}(X\Vert Y).$

The compact normalized $L^p$ norm is unchanged by conjugating a function and reflecting its argument: $\|x\mapsto\overline{f(-x)}\|_p=\|f\|_p$.

Hölder's inequality, binary case.

Translation preserves the compact normalized $L^p$ norm: for every group element $a$, $\|\tau_a f\|_{p}=\|f\|_{p}$.