_{\alpha }  f (x)& = \sup_{n > 0}   \Bigl\lvert \frac 1 { S_N} \sum_{n=1} ^{N} X_n f (x-n) \Bigr\rvert, \qquad S_N = \sum_{n=1} ^{N} X_n . \end{aligned}$$

Our sparse bound here is more restrictive, with the value of the sparse index $ r$ depending upon random parameter $ \alpha $.

\[t:R\] For any $ 0< \alpha < 1$, $ 1+ \alpha < r < 2$, almost surely, the following holds: For all functions $ f, g$ finitely supported on $ \mathbb Z $, there is a bilinear sparse operator $ \Lambda _{r} $ so that $$\lvert  \langle H _{\alpha } f, g \rangle\rvert  \lesssim \Lambda _{r}  (f,g).$$ The same inequality holds for $ M _{\alpha }$. (The sparse operator can be taken non-random, but the implied constant is random.)

Weighted inequalities are a corollary. They are *the first we know of* holding for operators defined on sets of the integers with zero asymptotic density.

\[c:R\] For any $ 0< \alpha < 1$, almost surely, the following holds: For all $ 1 + \alpha < p <  \frac {1+ \alpha } \alpha $, and weights $ w$ so that $$\label{e:WW}
w ^{1+ \alpha } \in A _{ (1+ \alpha ) (p-1)+1 