PFR

Let m be a positive integer. Suppose one has a sequence G_m → G_{m - 1} → ... → G_1 → G_0 = {0} of homomorphisms between abelian groups G_0, ...,G_m, and for each d=0, ...,m, let π_d : G_m → G_d be the homomorphism from G_m to G_d arising from this sequence by composition (so for instance π_m is the identity homomorphism and π_0 is the zero homomorphism). Let X_[m] = (X_1, ..., X_m) be a jointly independent tuple of G_m-valued random variables. Then D[X_[m]] = ∑ d, D[π_d(X_[m]) ,| , π_(d-1)(X_[m])] + ∑_{d=1}^{m - 1}, I[∑ i, X_i : π_d(X_[m]) | π_d(∑ i, X_i), π_(d-1})(X_[m])].

Under the preceding hypotheses, D[X_[m]] ≥ ∑ d, D[π_d(X_[m])| π_(d-1})(X_[m])] + I[∑ i, X_i : π_1(X_[m]) | π_1(∑ i, X_i)].

If $G$ is finite, then $\tau$ is continuous.

If $G$ is finite, then a $\tau$-minimizer exists.

If $G$ is finite, then a $\tau$-minimizer exists.

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

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

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

With the notation of the previous lemma, we have \begin{equation}\label{5.3-conv} k - D[ X'{[m]} | Y{[m]} ] \leq \eta \sum_{i=1}^m d[X_{\sigma(i)};X'_i|Y_i] \end{equation} for any permutation $\sigma : \{1,\dots,m\} \rightarrow \{1,\dots,m\}$.

If $H$ is a finite subgroup of $G$, then $\rho^-(U_H) = \log |A| - \log \max_t |A \cap (H+t)|$.

We have $\rho(U_A) = 0$.