  We study discrete random variants of the Carleson maximal operator.
Intriguingly, these questions remain subtle and difficult, even in this
setting. Let $\{X_m\}$ be an independent sequence of $\{0,1\}$ random variables
with expectations \[ \mathbb E X_m = \sigma_m = m^{-\alpha}, \ 0 < \alpha <
1/2, \] and $ S_m = \sum_{k=1} ^{m} X_k$. Then the maximal operator below
almost surely is bounded from $ \ell ^{p}$ to $ \ell ^{p}$, provided the
Minkowski dimension of $ \Lambda \subset [-1/2, 1/2]$ is strictly less than $
1- \alpha $. \[ \sup_{\lambda \in \Lambda } \Bigl| \sum_{m\neq 0} X_{\lvert
m\rvert } \frac{e( \lambda m )}{ {\rm sgn} (m)S_{ |m| }} f(x- m) \Bigr|. \]
This operator also satisfies a sparse type bound. The form of the sparse bound
immediately implies weighted estimates in all $ \ell ^{2}$, which are novel in
this setting. Variants and extensions are also considered.
