Research proof index

Among the first $n+1$ enumerated phases, enumΘ'ArgMax returns the smallest index $i$ at which the function $g$ attains its maximum at $x$. Equivalently, every index $j\le n$ has value at most the value at $i$, and every earlier index $j<i$ has strictly smaller value.

Any set of tiles can be written as the union of disjoint subfamilies, their number being controlled by the maximal stack size.

Let $g$ have bounded finite support, let $r>0$, and suppose $d(x,x')\le r$. The truncated Calderón-Zygmund operator changes by at most a constant times the global maximal function:

$d\bigl(T_r g(x),T_r g(x')\bigr)\le C(a)\,M g(x).$

Let $f$ be $2\pi$-periodic and belong to $L^q((0,2\pi])$ for some $q>1$. Given thresholds $\delta,\varepsilon>0$, there is an index $N_0$ such that the set where the tail error $\sup_{N>N_0}\lVert f(x)-S_Nf(x)\rVert$ exceeds $\delta$ has measure at most $\varepsilon$.

For a continuous, $2\pi$-periodic function $f$ and any $\delta,\varepsilon>0$, there is an index $N_0$ such that the set of $x\in(0,2\pi]$ for which $\sup_{N>N_0}\lVert f(x)-S_Nf(x)\rVert$ exceeds $\delta$ has measure at most $\varepsilon$.

If a tile $p$ lies in the $n$th minimal layer of a set of tiles $A$, then there is a tile $p'$ in the zeroth minimal layer with $p'\le p$, and the scale of $p$ is at least the scale of $p'$ plus $n$.