Research proof index

If $A \subset {\bf F}_2^n$ is finite non-empty with $|A+A| \leq K|A|$, then there exists a subgroup $H$ of ${\bf F}_2^n$ with $|H| \leq |A|$ such that $A$ can be covered by at most $2K^9$ translates of $H$.

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 $|H|\le|A|$.

If $X_1, X_2$ is a $\phi$-minimizer, then $d[X_1;X_2] = 0$.

Fix a nonempty finite set $A$ in an elementary abelian $2$-group. For any measurable random variables $Y_1,Y_2$, there are a subspace $H$ and a random variable $U$ uniformly distributed on $H$ such that the source's $\rho[\,\cdot\,\#A]$ functional satisfies $2\rho[U\#A]\le\rho[Y_1\#A]+\rho[Y_2\#A]+8d[Y_1;Y_2]$.