Research proof index

Multidistance is unchanged when a finite family of random variables is reindexed along an equality $m'=m$. This says that the quantity depends on the family, not on the particular equal presentation of its finite index type.

Let m ≥ 2, and let X_[m] be a tuple of G-valued random variables. Let W := ∑ X_i. Then d[W;-W] ≤ 2 D[X_i].

If $(X_i)_{1 \leq i \leq m}$ is a $\tau$-minimizer, then $\sum_{i=1}^m d[X_i; X^0] \leq \frac{2m}{\eta} d[X^0; X^0]$.

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 $|H|\le|A|$.