}, \qquad w   \in A _{1 + \frac 1 {(1+\alpha) (p'-1)}} ,$$ we have $
\lVert  H _{\alpha } \;:\; \ell ^{p} (w) \mapsto \ell ^{p} (w)\rVert  < \infty  
$. The implied constant only depends upon $ [ w ^{1+ \alpha }] _{A _{ (1+ \alpha ) (p-1)+1 }}$, and $ [w] _{A_{1 + \frac 1 {\alpha (p'-1)}} }$. The same inequality holds for $ M _{\alpha }$.

The study of these questions was initiated by Bourgain [@MR937581], as an elementary example of a sequence of integers for which one could derive $ \ell ^{p}$ inequalities, with the sequence of integers also having asymptotic density zero. Various aspects of these questions have been studied, both in $ \ell ^{p}$, in the weak $ (1,1)$ endpoints [@MR1325697; @MR2680392; @MR3421994; @MR2318621; @MR2576702]. We are not aware of any result in the literature that proves a weighted estimate in this sort of discrete setting. (If the set of integers has full density, it is easy to transfer weighted estimates.)

There is a subtle difference between the Hilbert transform and the maximal function in this random setting. In particular, more should be true for the maximal function. Prompted by the work of LaVictoire [@MR2576702], 