Research proof index

For every set $A$ in the ambient finite $\mathbb F_2$-vector space, some linear functional $\varphi:G\to\mathbb F_2$ has at least $(|A|-1)/2$ elements of $A$ in its $1$-fiber.

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 chain rule for conditional multidistance. Let $\pi:G\to H$ be a homomorphism, and suppose the pairs $(X_i,Y_i)$ are independent across the finite index set. Then $D[X\mid Y]=D[X\mid(\pi X,Y)]+D[\pi X\mid Y]+I\!\left[\sum_iX_i:(\pi X_i)_i\,\middle|\,\left(\pi\!\left(\sum_iX_i\right),(Y_i)_i\right)\right].$ The first term measures the remaining fiberwise multidistance after adjoining each image $\pi(X_i)$ to its conditioning data.

If $X, Y$ are $G$-valued random variables, and $Z$ is another random variable defined on the same sample space as $X$, then $D_{KL}((X|Z)\Vert Y) = D_{KL}(X\Vert Y) + \bbH[X] - \bbH[X|Z].$