Research proof index

Let $X_1',X_2'$ be independent copies of the $\tau$-minimizers $X_1,X_2$. Write $k=d[X_1;X_2]$ and $I_1=I[X_1+X_2:X_1'+X_2\mid X_1+X_2+X_1'+X_2']$. Then the entropy of the four-variable sum obeys $H[X_1+X_2+X_1'+X_2']\le\tfrac12H[X_1]+\tfrac12H[X_2]+(2+\eta)k-I_1$.

entropic_PFR_conjecture: For two $G$-valued random variables $X^0_1, X^0_2$, there is some subgroup $H \leq G$ such that $d[X^0_1;U_H] + d[X^0_2;U_H] \le 11 d[X^0_1;X^0_2]$.

In the project's entropic PFR package with parameter $\eta=1/9$, there is a subspace $H$ and a random variable $U$ uniformly distributed on $H$ such that each reference variable is within six times their mutual Ruzsa distance of $U$: $d(X_1,U),d(X_2,U)\le6d(X_1,X_2)$.

If $d[X_1;X_2]=0$, then there exists a subgroup $H \leq G$ such that $d[X_1;U_H] = d[X_2;U_H] = 0$. Follows from the preceding claim by the triangle inequality.

If $d[X ;X]=0$, then there exists a subgroup $H \leq G$ such that $d[X ;U_H] = 0$.

Let $H$ be a subgroup of $G \times G'$. Then there exists a subgroup $H_0$ of $G$, a subgroup $H_1$ of $G'$, and a homomorphism $\phi: G \to G'$ such that $H := \{ (x, \phi(x) + y): x \in H_0, y \in H_1 \}.$ In particular, $|H| = |H_0| |H_1|$.