Consider the following excerpt from this paper:
Given $1<p<2$, $0<w\leq 1$ and $n\in\mathbb{N}$, we fix once and for all a sequence $f_j^{(n)}=f_{p,w,j}^{(n)}$, $1\leq j\leq n$, of independent, symmetric, 3-valued random variables with $\|f_j^{(n)}\|_{L_p}=1$ and $\|f_j^{(n)}\|_{L_2}=\frac{1}{w}$ for $1\leq j\leq n$. We then define $F_{p,w}^{(n)}$ to be the subspace $\text{span}\{f_j^{(n)}:1\leq j\leq n\}$ of $L_p$. ... Since the $f_j^{(n)}$ are 3-valued, $F_{p,w}^{(n)}$ is a subspace of the span of indicator functions of $3^n$ pairwise disjoint sets. Thus, we can and will think of $F_{p,w}^{(n)}$ as a subspace of $\ell_p^{k_n}$, where $k_n=3^n$.
Question 1. Is there a reference to show that such $f_j^{(n)}$ exist? I guess we could use something like the Rademacher functions, except 3-valued. Given any $w\in(0,1]$ and any partition $[0,1]=A\cup B\cup C$ with Lebesgue measures $\lambda(A)=\lambda(C)=\frac{1}{2}w^{2p/(2-p)}$, we can always construct a symmetric 3-valued random variable $r\in L_p$ by letting \begin{equation*}r(t)=\left\{\begin{array}{ll}-w^{-2/(2-p)}&\text{ if }t\in A\\0&\text{ if }t\in B\\w^{-2/(2-p)}&\text{ if }t\in C,\end{array}\right.\end{equation*} and this satisfies $\|r\|_{L_p}=1$ and $\|r\|_{L_2}=\frac{1}{w}$. It shouldn't be hard to choose a sequence $(A_j,B_j,C_j)_{j=1}^n$ so that the corresponding sequence $(r_j)_{j=1}^n$ is independent. However, I would prefer to just cite a reference rather than build them from scratch.
Question 2. I do not understand the last sentence quoted above, about $F_{p,w}^{(n)}$ being viewed as a subspace of $\ell_p^{3^n}$. I get that $F_{p,w}^{(n)}$ is a subspace of the $L_p$-span of characteristic functions on $3^n$ pairwise disjoint subsets of $[0,1]$. However, why should this span be isometrically isomorphic to a subspace of $\ell_p^{3^n}$? Or even just uniformly isomorphic? (Although, if it is only uniformly isomorphic, then surely the specific constants in the paper must be adjusted to compensate.)
Then, on a somewhat unrelated note:
Question 3. The fact that independent symmetric 3-valued random variables in $L_p$ play such a special role was first (?) discussed by Rosenthal, who seemed to be motivated by the role of independent symmetric 2-valued random variables in $L_p$ in Khintchine's inequality. It is natural to conjecture that 4-valued ones will also have nice properties. Has this been studied?
