Research proof index

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 $B$.

For a tile $u$, the union of the grid cubes in its level-$n$ boundary family has measure at most a constant $C(X,n)$ times the measure of the spatial cube $\mathcal{I}(u)$.

The real-line Carleson operator is positively homogeneous. For every $a>0$,

$T f(x)=a\,T(f/a)(x),$

where the scalar on the right is interpreted in the extended nonnegative reals.

Chang's lemma for the large Fourier spectrum. If $f$ is nonzero and $\eta>0$, there is a subset $\Delta$ of the $\eta$-large spectrum such that the entire large spectrum lies in the additive span of $\Delta$. The theorem also gives the explicit bound $|\Delta| \le \left\lceil C e\,\left\lceil \mathcal L\!\left(\|f\|_1^2/(\|f\|_2^2|G|)\right)\right\rceil/\eta^2\right\rceil$, with the project's constant $C$.