Research proof index

Two 2×2 self-adjoint matrices are equal if the (complex) traces of each matrix multiplied by each of the Pauli-matrices are equal.

The integral appearing in corrugations is periodic.

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|.

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 $|H|\le|A|$.

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 $A\subseteq c+H$, $|c|<2K^{12}$, and $|H|\le|A|$.