Research proof index

The polynomial Freiman-Ruzsa (PFR) conjecture: if A is a subset of an elementary abelian 2-group of doubling constant at most K, then A can be covered by at most 2 * K ^ 12 cosets of a subgroup of cardinality at most |A|.

Improved polynomial Freiman-Ruzsa theorem. Let $A$ be a nonempty subset of a finite elementary abelian $2$-group. If $|A+A|\le K|A|$, then there are a subspace $H$ and a set of representatives $c$ such that $A\subseteq c+H$, $|c|<2K^{11}$, and $|H|\le|A|$.

Improved 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 $A\subseteq c+H$, $|c|<2K^{11}$, and $|H|\le|A|$.

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 $|H|\le|A|$.

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 $d[X_1+X_2';X_2+X_1']+d[X_1\mid X_1+X_2';X_2\mid X_2+X_1']+I[X_1+X_2:X_1'+X_2\mid X_1+X_2+X_1'+X_2']=2k$.