Research proof index

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 $\pi/2$: $B_{\mathrm{chord}}\subseteq ((\pi/2)B)_{\mathrm{arc}}$.

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.