positive constants (also when they appear in a sequence of relations).

\[pro:Q1\] There is a constant $c>0$ such that for all $n,N\in {{\mathbb N}}$ and $g\in L_1^N$ $$e_n^{{\rm q }}(S_{N,g},B_{L_\infty^N})\le cn^{-1}\|g\|_{ L_1^N}.$$

First we consider the case ${{\mathbb K}}={{\mathbb R}}$. If $g=0$, the statement is trivial. We may assume without loss of generality that $g\ge 0$, otherwise we split $g$ into its positive and negative part and apply Proposition \[pro:3\]. Moreover, by scaling the problem appropriately, we can assume $$\label{QC4}
\|g\|_{L_1^N}=1.$$ Now we reduce the problem $S_{N,g}$ to the known case $S_M$ for some $M$. Define $h,\tilde{g}\in L_1^N$ by $$\label{QC5}
h(i)=\lfloor ng(i)\rfloor, \quad \tilde{g}(i)=n^{-1}h(i)\quad(i=0,\dots,N-1).$$ We have $|g(i)-\tilde{g}(i)|\le n^{-1}$, therefore $$\sup_{f\in B_{L_\infty^N}}|S_{N,g}f-S_{N,\tilde{g}}f|\le n^{-1}.$$ By Lemma 6 of [@Hei01], $$\label{QC6}
e_n^{{\rm q }}(S_{N,g},B_{L_\infty^N})\le e_n^{{\rm q }}(S_{N,\tilde{g}},B_{L_\infty^N})+n^{-1}.$$ Now set $m_0=0$ and for $1\le i\le N$ $$m_i=\sum_{l=0}^{i-1} h(l),$$ and denote $m_N=M$. The case $M=0$ is trivial, since this implies $\tilde{g}\equiv 0$, thus the result follows directly from (\[QC6\]). Hence we assume $M\ge 1$. Observe that (\[QC4\]) and (\[QC5\]) imply $$\label{QC7}
M\le nN.$$ Define $$\eta:{{\mathbb Z}}[0,M)\to {{\mathbb Z}}[0,N)$$ by $\eta(j)=i$, where $i$ is the unique integer satisfying $m_i\le j< m_{i+1}$. Let the reduction mapping $R:L_\infty^N\to L_\infty^M$ be given by $$(R(f))(j)=f(\eta(j))\quad (j=0,\dots,M-1).$$ Clearly, $R$ is of 