Research proof index

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 $I( Z_1 - Z_2 : (\pi(Z_1), \pi(Z_2)) | \pi(Z_1 - Z_2) ).$

A Ruzsa-distance lower bound for sums of independent copies. Let $X_1',X_2'$ be independent copies of the $\tau$-minimizers $X_1,X_2$, and set $k=d[X_1;X_2]$. Then $d[X_1+X_1';X_2+X_2']\ge k-\frac{\eta}{2}\bigl(d[X_1;X_1]+d[X_2;X_2]\bigr).$

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 $2\rho[U\#A]\le\rho[Y_1\#A]+\rho[Y_2\#A]+8d[Y_1;Y_2]$.

The second information estimate for $\tau$-minimizers. Let $X_1',X_2'$ be independent copies of $X_1,X_2$, set $k=d[X_1;X_2]$, $I_1=I[X_1+X_2:X_1'+X_2\mid X_1+X_2+X_1'+X_2']$, and $I_2=I[X_1+X_2:X_1'+X_1\mid X_1+X_2+X_1'+X_2']$. Then $I_2\le2\eta k+\dfrac{2\eta(2\eta k-I_1)}{1-\eta}$.

If $(X_i)_{1 \leq i \leq m}$ is a $\tau$-minimizer, and $k := D[(X_i)_{1 \leq i \leq m}]$, then for any other tuples $(X'_i)_{1 \leq i \leq m}$ and $(Y_i)_{1 \leq i \leq m}$ with the $X'_i$ G$-valued, one has $k - D[(X'_i)_{1 \leq i \leq m} | (Y_i)_{1 \leq i \leq m}] \leq \eta \sum_{i=1}^m d[X_i; X'_i|Y_i].$