Research proof index

Alice produces (p,y) with p close to o.probs y with good probability

Alice produces (p,y) with p close and y true with good probability, since if we condition on Alice being close she does as least as well as a close oracle.

An $L^\infty$ almost-periodicity theorem. Under the small-growth hypothesis $\sigma[A,S]\le K$, and for nonempty finite sets $B,C$, there is a set of translations $T$ with $|T|\ge K^{-4096\lceil\mathcal L(|C|/|B|)\rceil/\varepsilon^2}|S|$. Every $t\in T$ changes the normalized convolution $\mu_A*1_B*\mu_C$ by at most $\varepsilon$ in $L^\infty$.

A boosted $L^\infty$ almost-periodicity estimate. Under $\sigma[A,S]\le K$, it finds a large set $T$, with the stated lower bound $|T|\ge K^{-4096\lceil\mathcal L(|C|/|B|)\rceil k^2/\varepsilon^2}|S|$, such that averaging the target convolution against the $k$-fold convolution of $\mu_T$ changes it by at most $\varepsilon$ in $L^\infty$.

Let $f$ be analytic on an open set $s$ containing $0$, and suppose $f(0)=0$. Define $g(z)=f(z)/z$ for $z\ne0$ and $g(0)=f'(0)$. Then the apparent singularity at $0$ is removable and $g$ is analytic throughout $s$.

Suppose $R>0$ and $f$ is analytic on the closed disc $|z|\le R$ with $f(0)=0$. Define $g(z)=f(z)/z$ for $z\ne0$ and $g(0)=f'(0)$. Then $g$ is analytic on the entire closed disc, including at the removed singularity.