Research proof index

If $d[X_1;X_2]=0$, then there exists a subgroup $H \leq G$ such that $d[X_1;U_H] = d[X_2;U_H] = 0$. Follows from the preceding claim by the triangle inequality.

If $d[X ;X]=0$, then there exists a subgroup $H \leq G$ such that $d[X ;U_H] = 0$.

Let $V$ be the intersection of the kernels of a set of additive characters $\Delta$. Averaging $f$ over $V$ equals the average of $\widehat f(\psi)$ over all characters, with the Fourier coefficient retained precisely when $\psi$ lies in the additive subgroup generated by $\Delta$.

Let $G$ be a finite vector space over $\mathbb F_q$, where $q\ge3$ is prime. If a nonempty set $A\subseteq G$ contains no nontrivial three-term arithmetic progression, then $\dim_{\mathbb F_q}G \le 2^{148}\,\mathcal L(\alpha)^9$, where $\alpha=|A|/|G|$ and $\mathcal L$ is the project's capped logarithm.

A weighted Hölder lower bound for Fourier energy. If $\Delta$ lies in the $\eta$-large spectrum of $f$, $m\ne0$, and a weight $\nu$ is at least $1$ wherever $f$ is nonzero, then the order-$m$ energy of $\Delta$ weighted by $\widehat\nu$ is at least $|\Delta|^{2m}\eta^{2m}\|f\|_1^2/\|f\|_2^2$.

Let $H$ be a subgroup of $G \times G'$. Then there exists a subgroup $H_0$ of $G$, a subgroup $H_1$ of $G'$, and a homomorphism $\phi: G \to G'$ such that $H := \{ (x, \phi(x) + y): x \in H_0, y \in H_1 \}.$ In particular, $|H| = |H_0| |H_1|$.