Inequalities for the distribution of a sum of functions of independent random variables

Let $\xi=\sum_{i_1,\dots,i_r=1}^nf_{i_1,\dots,i_r=1}(\zeta_{i_1,\dots,i_r=1})$ where $\zeta_1,\dots,\zeta_n$ are independent random variables and the $f_{i_1,\dots,i_r=1}$ are functions (e.g., taking the values 0 and 1). For cases when “almost all” the summands forming $\xi$ are equal to 0 with a probability close to 1, estimates from above and below are obtained for the quantity $\mathsf P\{\xi=0\}$, as well as upper estimates for the distance in variation between the distribution $\xi$, and the distribution of the “approximating” sum of independent random variables.