Research proof index

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

A dependent-random-choice corollary. Let $A$ be nonempty, let $0<\varepsilon\le 1$ and $\delta>0$, and let $p$ be a nonzero even integer satisfying $\varepsilon^{-1}\log(2/\delta)\le p$. Then there are sets $A_1,A_2$ such that the normalized difference distribution $\mu_{A_1}\mathbin{\circleddash}\mu_{A_2}$ assigns mass at least $1-\delta$ to the source's sifted set $s_{p,\varepsilon}(A)$. Both sets retain explicit density: $\operatorname{dens}(A_i)\ge \tfrac14\operatorname{dens}(A)^{2p}$ for $i=1,2$.

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 $k - D[(X'_i)_{1 \leq i \leq m} | (Y_i)_{1 \leq i \leq m}] \leq \eta \sum_{i=1}^m d[X_i; X'_i|Y_i].$

In the $\tau$-minimizer endgame, let $X_1',X_2'$ be independent copies of $X_1,X_2$, set $k=d[X_1;X_2]$, $U=X_1+X_2$, $V=X_1'+X_2$, $W=X_1'+X_1$, $S=X_1+X_2+X_1'+X_2'$, and $I_1=I[U:V\mid S]$. If $c[A\mid S\#A\mid S]=\sum_{i=1}^2\bigl(d[X_i^0;A\mid S]-d[X_i^0;X_i]\bigr)$, then $c[U\mid S\#U\mid S]+c[V\mid S\#V\mid S]+c[W\mid S\#W\mid S]\le(6-3\eta)k+3(2\eta k-I_1)$.

Let $Y_1,Y_2,Y_3$ and $Y_4$ be independent $G$-valued random variables. Then $d[Y_1-Y_3; Y_2-Y_4] + d[Y_1|Y_1-Y_3; Y_2|Y_2-Y_4]$ $+ I[Y_1-Y_2 : Y_2 - Y_4 | Y_1-Y_2-Y_3+Y_4] = d[Y_1; Y_2] + d[Y_3; Y_4].$

For four measurable random variables $Y_0,Y_1,Y_2,Y_3$ in a finite abelian group, the conditional mutual-information term used in the fibring identity can be reduced to $I\!\left[(Y_0-Y_1,Y_2-Y_3):(Y_0-Y_2,Y_1-Y_3)\,\middle|\,Y_0-Y_1-Y_2+Y_3\right]=I[Y_0-Y_1:Y_1-Y_3\mid Y_0-Y_1-Y_2+Y_3].$