Polynomial Freiman-Ruzsa project

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 $x$.

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 $|H|\le|A|$.

In the ambient finite $\mathbb F_2$-vector space, exactly half of the additive homomorphisms $\varphi:G\to\mathbb F_2$ take a fixed nonzero vector $x$ to $1$: $2\,|\{\varphi:\varphi(x)=1\}|=|G|$.

For every set $A$ in the ambient finite $\mathbb F_2$-vector space, some linear functional $\varphi:G\to\mathbb F_2$ has at least $(|A|-1)/2$ elements of $A$ in its $1$-fiber.

A chain rule for conditional multidistance. Let $\pi:G\to H$ be a homomorphism, and suppose the pairs $(X_i,Y_i)$ are independent across the finite index set. Then $D[X\mid Y]=D[X\mid(\pi X,Y)]+D[\pi X\mid Y]+I\!\left[\sum_iX_i:(\pi X_i)_i\,\middle|\,\left(\pi\!\left(\sum_iX_i\right),(Y_i)_i\right)\right].$ The first term measures the remaining fiberwise multidistance after adjoining each image $\pi(X_i)$ to its conditioning data.

If $X, Y$ are $G$-valued random variables, and $Z$ is another random variable defined on the same sample space as $X$, then $D_{KL}((X|Z)\Vert Y) = D_{KL}(X\Vert Y) + \bbH[X] - \bbH[X|Z].$

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.

A lower bound forced by $\tau$-minimality. If $(X_1,X_2)$ minimizes the source's $\tau$ functional, then for measurable $X_1',X_2'$ and conditioning variables $Z,W$, $d[X_1'\mid Z;X_2'\mid W]\ge d[X_1;X_2]-\eta\bigl(d[X_1^0;X_1'\mid Z]-d[X_1^0;X_1]\bigr)-\eta\bigl(d[X_2^0;X_2'\mid W]-d[X_2^0;X_2]\bigr).$