Anti-concentration for polynomials of independent random variables

We prove anti-concentration results for polynomials of independent random variables with arbitrary degree. Our results extend the classical Littlewood-Offord result for linear polynomials, and improve several earlier estimates. We discuss applications in two different areas. In complexity theory, we prove nearoptimal lower bounds for computing the PARITY function, addressing a challenge in complexity theory posed by Razborov and Viola, and also address a problem concerning the OR function. In random graph theory, we derive a general anti-concentration result on the number of copies of a fixed graph in a random graph.