Research proof index

The Beroulli case of Hoeffding's lemma

If $\varphi$ is supported in the ball $B(z,R)$, then the oscillatory integral with phase difference $f-g$ satisfies

$\left\lVert\int e^{i(f(x)-g(x))}\varphi(x)\,dx\right\rVert \le C(a)\,\mu(B(z,R))\,\lVert\varphi\rVert_{\mathrm{Hol},\tau;B(z,2R)}\,(1+d_{z,R}(f,g))^{-1/(2a^2+a^3)}.$

If the project’s inhomogeneous $t$-Hölder norm of $\varphi$ on the ball $B(z,R)$ is finite and $t\ge0$, then $\varphi$ is $t$-Hölder on that ball. A valid Hölder constant is the finite normalized norm divided by $R^t$.

Let $f: G \to G'$ be a function, and let $S$ denote the set $S := \{ f(x+y)-f(x)-f(y): x,y \in G \}.$ Then there exists a homomorphism $\phi: G \to G'$ such that $|\{f(x) - \phi(x)\}| \leq |S|^{10}.$

Two maps are equal if they agree on all positive inputs with trace one

Shannon entropy of a distribution is at most ln d.