Bilinear and quadratic variants on the Littlewood-Offord problem

If f(x (1), aEuro broken vertical bar, x (n) ) is a polynomial dependent on a large number of independent Bernoulli random variables, what can be said about the maximum concentration of f on any single value? For linear polynomials, this reduces to one version of the classical Littlewood-Offord problem: Given nonzero constants a (1), aEuro broken vertical bar,a (n) , what is the maximum number of sums of the form +/- a (1) +/- a (2) +/- aEuro broken vertical bar +/- a (n) which take on any single value? Here we consider the case where f is either a bilinear form or a quadratic form. For the bilinear case, we show that the only forms having concentration significantly larger than n (-1) are those which are in a certain sense very close to being degenerate. For the quadratic case, we show that no form having many nonzero coefficients has concentration significantly larger than n (-1/2). In both cases the results are nearly tight.