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 $|H|\le|A|$.

For a finite ambient group, the chord model of a Bohr set $B$ is contained in the arc model after widening $B$ by the factor $\pi/2$: $B_{\mathrm{chord}}\subseteq ((\pi/2)B)_{\mathrm{arc}}$.

Characterization of the order on Bohr sets. The relation $B_1\le B_2$ holds exactly when every frequency of $B_2$ is also a frequency of $B_1$, and $\operatorname{width}_{B_1}(\psi)\le \operatorname{width}_{B_2}(\psi)$ for each such frequency.

A point $x$ belongs to the arc model of a Bohr set $B$ exactly when every frequency $\psi$ of $B$ satisfies $\|\operatorname{angle}(\psi(x),1)\|\le \operatorname{width}_B(\psi)$. Thus membership can be checked using only the stored frequencies and widths.

A point $x$ belongs to the chord model of a Bohr set $B$ exactly when $\|1-\psi(x)\|\le \operatorname{width}_B(\psi)$ for every frequency $\psi$ of $B$.

In the ambient finite $\mathbb F_2$-vector space, exactly half of the additive homomorphisms $\varphi:G\to\mathbb F_2$ take a fixed nonzero vector $x$ to $1$: $2\,|\{\varphi:\varphi(x)=1\}|=|G|$.