Sparse Bounds for the Discrete Cubic Hilbert Transform

Consider the discrete cubic Hilbert transform defined on finitely supported functions $f$ on $mathbb{Z}$ by begin{eqnarray*} H_3f(n) = sum_{m ot = 0} frac{f(n- m^3)}{m}. end{eqnarray*} We prove that there exists $r <2$ and universal constant $C$ such that for all finitely supported $f,g$ on $mathbb{Z}$ there exists an $(r,r)$-sparse form ${Lambda}_{r,r}$ for which begin{eqnarray*} left| langle H_3f, g rangle right| leq C {Lambda}_{r,r} (f,g). end{eqnarray*} This is the first result of this type concerning discrete harmonic analytic operators. It immediately implies some weighted inequalities, which are also new in this setting.