Research proof index

Improved polynomial Freiman-Ruzsa theorem. Let $A$ be a nonempty subset of a finite elementary abelian $2$-group. If $|A+A|\le K|A|$, then there are a subspace $H$ and a set of representatives $c$ such that $A\subseteq c+H$, $|c|<2K^{11}$, and $|H|\le|A|$.

Improved polynomial Freiman-Ruzsa theorem 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 $A\subseteq c+H$, $|c|<2K^{11}$, and $|H|\le|A|$.

For p.η ≤ 1/8, there exist τ-minimizers X₁, X₂ at zero Rusza distance. For p.η < 1/8, all minimizers are fine, by tau_strictly_decreases'. For p.η = 1/8, we use a limit of minimizers for η < 1/8, which exists by compactness.