Research proof index

Rudin's inequality, exponential form.

Rudin's inequality, usual form.

A dependent-random-choice corollary. Let $A$ be nonempty, let $0<\varepsilon\le 1$ and $\delta>0$, and let $p$ be a nonzero even integer satisfying $\varepsilon^{-1}\log(2/\delta)\le p$. Then there are sets $A_1,A_2$ such that the normalized difference distribution $\mu_{A_1}\mathbin{\circleddash}\mu_{A_2}$ assigns mass at least $1-\delta$ to the source's sifted set $s_{p,\varepsilon}(A)$. Both sets retain explicit density: $\operatorname{dens}(A_i)\ge \tfrac14\operatorname{dens}(A)^{2p}$ for $i=1,2$.

For a finite group of odd order and a three-term-progression-free set $s$, the normalized inner product between $\mu_s*\mu_s$ and the uniform measure on $2s$ is exactly $|s|^{-2}$. The identity records the precise normalized count forced by the absence of nontrivial three-term progressions.

An unbalancing lemma in physical space. Suppose $\nu$ is a probability weight, $f$ is real-valued, and $f$ and $\nu$ admit self-difference-convolution factorizations $f=g\mathbin{\circleddash}g$ and $\nu=h\mathbin{\circleddash}h$. If $0<\varepsilon\le1$, $p\ne0$, and $\|f\|_{L^p(\nu)}\ge\varepsilon$, then some integer $p'$ satisfies $p'\le 2^{10}\varepsilon^{-2}p$ and $\|1+f\|_{L^{p'}(\nu)}\ge1+\varepsilon/2$.

Parseval-Plancherel identity for the discrete Fourier transform.