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 $U$: $d(X_1,U),d(X_2,U)\le6d(X_1,X_2)$.

Apply the optional stopping theorem to get equation 4. Note that T1 Space is needed to make sure that mesh ι n has order topology.

Evil Alice produces a close true trace with low probability, since by remaining close she looks like a close oracle.

If Alice is good, the probability of false is low