Sparse Bounds for Discrete Quadratic Phase Hilbert Transform

Consider the discrete quadratic phase Hilbert Transform acting on $ell^{2}$ finitely supported functions $$ H^{alpha} f(n) : = sum_{m eq 0} frac{e^{2 pi ialpha m^2} f(n - m)}{m}. $$ We prove that, uniformly in $alpha in mathbb{T}$, there is a sparse bound for the bilinear form $langle H^{alpha} f , g rangle$. The sparse bound implies several mapping properties such as weighted inequalities in an intersection of Muckenhoupt and reverse Holder classes.