Research proof index

If Alice is good, the probability of false is low

If Alice is correct and Bob rejects, the probability of false is low

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 $d[X_1;X_2]=0$, then there exists a subgroup $H \leq G$ such that $d[X_1;U_H] = d[X_2;U_H] = 0$. Follows from the preceding claim by the triangle inequality.

If $d[X ;X]=0$, then there exists a subgroup $H \leq G$ such that $d[X ;U_H] = 0$.