Research proof index

A special case of Young's convolution inequality.

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

A dependent-random-choice estimate. For $p \ge 2$, a nonnegative function $f$, nonempty $A$, and intersecting sets $B_1,B_2$, the support hypothesis produces subsets $A_1 \subseteq B_1$ and $A_2 \subseteq B_2$ whose normalized difference convolution has controlled correlation with $f$. Both relative sizes $|A_i|/|B_i|$ are bounded below by the same explicit quantity, namely one quarter of a normalized $2p$-th power of the weighted $L^p$ norm of $1_A \mathbin{\circleddash} 1_A$.

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 $|\Delta|^{2m}\eta^{2m}\|f\|_1^2/\|f\|_2^2$.