Research proof index

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 $d[X_1+X_1';X_2+X_2']\ge k-\frac{\eta}{2}\bigl(d[X_1;X_1]+d[X_2;X_2]\bigr).$

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 $I_2\le2\eta k+\dfrac{2\eta(2\eta k-I_1)}{1-\eta}$.