A sparse quadratic T(1) theorem

We show that any Littlewood--Paley square function S satisfying a minimal Carleson condition is dominated by a sparse form. This implies strong weighted Lp estimates for all Ap weights with sharp dependence on the Ap characteristic. In particular, the Carleson condition and the sparse domination are equivalent. The proof uses random dyadic grids, decomposition in the Haar basis, and a stopping time argument.