  We prove several different anti-concentration inequalities for functions of
independent Bernoulli-distributed random variables. First, motivated by a
conjecture of Alon, Hefetz, Krivelevich and Tyomkyn, we prove some
"Poisson-type" anti-concentration theorems that give bounds of the form 1/e +
o(1) for the point probabilities of certain polynomials. Second, we prove an
anti-concentration inequality for polynomials with nonnegative coefficients
which extends the classical Erd\H{o}s-Littlewood-Offord theorem and improves a
theorem of Meka, Nguyen and Vu for polynomials of this type. As an application,
we prove some new anti-concentration bounds for subgraph counts in random
graphs.
