Research proof index

A point $x$ belongs to the arc model of a Bohr set $B$ exactly when every frequency $\psi$ of $B$ satisfies $\|\operatorname{angle}(\psi(x),1)\|\le \operatorname{width}_B(\psi)$. Thus membership can be checked using only the stored frequencies and widths.

A point $x$ belongs to the chord model of a Bohr set $B$ exactly when $\|1-\psi(x)\|\le \operatorname{width}_B(\psi)$ for every frequency $\psi$ of $B$.

A closed-disc form of the Borel–Carathéodory theorem. Let $R,M>0$, let $f$ be analytic on $|z|\le R$, assume $f(0)=0$, and suppose $\operatorname{Re}f(z)\le M$ throughout that disc. If $r<R$ and $|z|\le r$, then $|f(z)|\le \frac{2Mr}{R-r}.$

For a tile $u$, the union of the grid cubes in its level-$n$ boundary family has measure at most a constant $C(X,n)$ times the measure of the spatial cube $\mathcal{I}(u)$.

For the Selberg sieve that removes primes up to $z$ from an interval $[x,x+y]$, with $x,y>0$ and $z\ge1$, the remainder in the expected count of multiples of every nonzero integer $d$ is uniformly bounded: $|r_d|\le5$.

Suppose sequences $f,g:\mathbb N\to\mathbb R$ satisfy $f(n)=O(g(n))$ as $n\to\infty$, and $f(n)=0$ whenever $g(n)=0$. Then a single constant bounds $|f(n)|$ by that constant times $|g(n)|$ for every natural number $n$, not only for all sufficiently large $n$.