Research proof index

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 $B$.

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 $C$.

For a complex-valued function on the ambient finite group and a nonzero even integer $n$, ordinary self-convolution has no larger normalized $L^n$ norm than self-difference-convolution: $\|f*f\|_n\le\|f\mathbin{\circleddash}f\|_n$.

The $2n$-th Fourier moment of the indicator of a finite set equals its order-$n$ additive energy: $\|\widehat{1_s}\|_{2n}^{2n}=E_n(s)$. This is the standard bridge between Fourier norms and additive tuple counts.

A finite-field density-increment lemma. If the normalized additive correlation of $A$ with a set $C$ of density at least $\gamma$ differs from its random value by at least $\varepsilon$, then there is a subspace $V$ of explicitly bounded codimension. Averaging $1_A$ over $V$ raises its $L^\infty$ density to at least $(1+\varepsilon/32)\alpha$, where $\alpha$ is the density of $A$.