Research proof index

Conditional multidistance is unchanged when both the random variables and their conditioning variables are reindexed along an equality $m'=m$. As with ordinary multidistance, the value does not depend on the chosen equal presentation of the finite index type.

A lower bound forced by $\tau$-minimality. If $(X_1,X_2)$ minimizes the source's $\tau$ functional, then for measurable $X_1',X_2'$ and conditioning variables $Z,W$, $d[X_1'\mid Z;X_2'\mid W]\ge d[X_1;X_2]-\eta\bigl(d[X_1^0;X_1'\mid Z]-d[X_1^0;X_1]\bigr)-\eta\bigl(d[X_2^0;X_2'\mid W]-d[X_2^0;X_2]\bigr).$

A preliminary endgame bound. In the $\tau$-minimizer setup, let $k=d[X_1;X_2]$ and let measurable $T_1,T_2,T_3$ satisfy $T_1+T_2+T_3=0$. Put $\delta=I[T_1:T_2]+I[T_2:T_3]+I[T_3:T_1]$ and $c[T_1\#T_2]=d[X_1^0;T_1]-d[X_1^0;X_1]+d[X_2^0;T_2]-d[X_2^0;X_2].$ Then $k\le\delta+\eta c[T_1\#T_2]+\frac{\eta}{2}\bigl(I[T_1:T_3]+I[T_2:T_3]\bigr).$

A finite-field density-increment lemma. If the normalized additive correlation of $A$ with a set $C$ of density at least $\gamma$ differs from its random value by at least $\varepsilon$, then there is a subspace $V$ of explicitly bounded codimension. Averaging $1_A$ over $V$ raises its $L^\infty$ density to at least $(1+\varepsilon/32)\alpha$, where $\alpha$ is the density of $A$.

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

Let $T_1,T_2,T_3$ be measurable random variables in a finite abelian group with $T_1+T_2+T_3=0$, and set $\delta=I[T_1:T_2]+I[T_1:T_3]+I[T_2:T_3]$. For any measurable $Y_1,\ldots,Y_n$ and any $\alpha>0$, there is a measurable random variable $U$ such that $d[U;U]+\alpha\sum_{i=1}^n d[Y_i;U]\le\left(2+\frac{\alpha n}{2}\right)\delta+\alpha\sum_{i=1}^n d[Y_i;T_2].$