Joint boundedness of families of random Fourier series Let $\varepsilon_n$, $ n \in \mathbb {Z}$, be independent Rademacher random variables.
Is there a characterization of those sequences $a_{m,n}$, $(m,n) \in \mathbb {Z} \times \mathbb{Z}$, of complex numbers such that
$$\sup_{m \in \mathbb{Z}} \sup_{t \in [0,2\pi]}
 \bigg | \sum_{n \in \mathbb{Z}} a_{m,n} \varepsilon_n e^{int} \bigg| < \infty
$$
almost everywhere?
 A: $\newcommand{\Z}{\mathbb Z}\newcommand{\ep}{\varepsilon}\newcommand{\si}{\sigma}\newcommand{\la}{\lambda}\newcommand{\ga}{\gamma}$Let $b_n^{m,u}:=\Re(a_{m,n}e^{inu})$ and $c_n^{m,u}:=\Im(a_{m,n}e^{inu})$.
We want to have a necessary and sufficient condition for
\begin{equation*}
    |S|<\infty, \tag{1}
\end{equation*}
where
\begin{equation*}
    S=\sup_{t\in T}\sum_{n\in\Z}t_n\ep_n  
\end{equation*}
and
\begin{equation*}
    T:=\{(b_n^{m,u})_{n\in\Z}\colon m\in\Z,u\in[0,2\pi]\}; 
\end{equation*}
and similarly with $c_n^{m,u}$ in place of $b_n^{m,u}$.
By Lemma 6 in Section 2 of Ch. IX, (1) implies
\begin{equation*}
    \si:=\sup_{t\in T}\|t\|_2<\infty, \tag{2}
\end{equation*}
where $\|t\|_p:=(\sum_{n\in\Z}|t_n|^p)^{1/p}$.
The key result here is the solution by Bednorz and Latała of the so-called Bernoulli Conjecture. Indeed, by Theorem 1.1 and formula (1) of Bednorz and Latała, there is a  universal positive real constant $C$ such that
\begin{equation*}
    \la(T)/C\le ES\le C\la(T),
\end{equation*}
where
\begin{equation*}
    \la(T):=\inf\{\sup_{t\in T_1}\|t\|_1+\ga_2(T_2)\colon T\subseteq T_1+T_2, T_1\subseteq\ell^2(\Z), T_2\subseteq\ell^2(\Z)\}
\end{equation*}
and $\ga_2(T_2):=\ga_2(T_2,d)$ is as in formula (1) of Bednorz and Latała.
Moreover, by Theorem 2.5 of Bednorz and Latała and in view of (2), (1) implies $E|S|<\infty$.
Thus, (1) holds if and only if $\la(T)<\infty$.
