PFR

The distribution of X + Z is the convolution of the distributions of Z and X when these random variables are independent. Probably already available somewhere in some form, but I couldn't locate it.

If $X, Y, Z$ are independent $G$-valued random variables, then $D_{KL}(X+Z\Vert Y+Z) \leq D_{KL}(X\Vert Y).$

If $X, Y$ are $G$-valued random variables, and $Z$ is another random variable defined on the same sample space as $X$, then $D_{KL}((X|Z)\Vert Y) = D_{KL}(X\Vert Y) + \bbH[X] - \bbH[X|Z].$

Given two independent random variables U and V uniformly distributed respectively on A and B, then U = V with probability #(A ∩ B) / #A ⬝ #B.

Given two independent random variables U and V uniformly distributed respectively on A and B, then U = V + x with probability # (A ∩ (B + x)) / #A ⬝ #B.

A uniform distribution on a set with doubling constant K has self Rusza distance at most log K.

The polynomial Freiman-Ruzsa (PFR) conjecture: if A is a subset of an elementary abelian 2-group of doubling constant at most K, then A can be covered by at most 2 * K ^ 12 cosets of a subgroup of cardinality at most |A|.

Corollary of PFR_conjecture in which the ambient group is not required to be finite (but) then H and c are finite.

If $S$ is a finite set, and $a_s$, $b_s$ are non-negative for $s\in S$, then

If equality holds in the previous bound, then $a_s=r\cdot b_s$ for every $s\in S$, for some constant $r\in \mathbb{R}$.

Given a submodule $E$ of $B \times F$, there is an equivalence $f : E \to B' \times F'$ given by the projections $E \to B$ and $E \to F$ "modulo" some $φ : B \to F$.

Probability measures on a finite space tend to a limit if and only if the probability masses of all points tend to the corresponding limits. Version in ℝ≥0.

The law of $(X, Z)$ is the image of the law of $(Z,X)$.

For X, Y random variables, one can find independent copies X', Y' of X, Y.

For X, Y random variables, one can find independent copies X', Y' of X, Y. Version formulated in spaces with a canonical measures.