Research proof index

Polynomial Freiman-Ruzsa theorem with exponent $9$, without a finite ambient-group assumption. Let $A$ be a nonempty finite subset of an elementary abelian $2$-group. If $|A+A|\le K|A|$, then there are a finite subspace $H$ and a finite set $c$ such that $A\subseteq c+H$, $|c|<2K^9$, and $|H|\le|A|$.

Honest Bob usually accepts if Alice is off by ≤ c

Bob makes few queries, regardless of Alice and Vera

Honest Bob usually rejects if Alice is off by ≥ s

Bob makes few queries, regardless of Alice and Vera

If Alice lies about probabilities by more than b, Bob usually catches Alice in a lie