Research proof index

Polynomial Freiman-Ruzsa theorem 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 $A\subseteq c+H$, $|c|<2K^{12}$, and $|H|\le|A|$.

A positivity lemma for self-difference-convolutions. If $f=g\mathbin{\circleddash}g$ and the nonnegative weight $\nu$ has a factorization $\nu=h\mathbin{\circleddash}h$, then every natural power of $f$ has nonnegative weighted inner product with $\nu$: $\langle f^k,\nu\rangle\ge0$ for every $k\in\mathbb N$.

Two internally independent families remain jointly independent after they are combined over a disjoint union, provided the two family-valued random variables are independent of one another.

A fibring identity in the $\tau$-minimizer setup. Let $X_1',X_2'$ be independent copies of $X_1,X_2$ and put $k=d[X_1;X_2]$. Then $d[X_1+X_2';X_2+X_1']+d[X_1\mid X_1+X_2';X_2\mid X_2+X_1']+I[X_1+X_2:X_1'+X_2\mid X_1+X_2+X_1'+X_2']=2k$.

A uniform distribution on a set with doubling constant K has self Rusza distance at most log K.

If $Z_1, Z_2$ are independent, then $d[Z_1; Z_2]$ is equal to $d[\pi(Z_1);\pi(Z_2)] + d[Z_1|\pi(Z_1); Z_2 |\pi(Z_2)]$ plus $I( Z_1 - Z_2 : (\pi(Z_1), \pi(Z_2)) | \pi(Z_1 - Z_2) ).$