  Consider a quadratic polynomial $Q(\xi_{1},\dots,\xi_{n})$ of independent
Rademacher random variables $\xi_{1},\dots,\xi_{n}$. To what extent can
$Q(\xi_{1},\dots,\xi_{n})$ concentrate on a single value? This quadratic
version of the classical Littlewood--Offord problem was popularised by
Costello, Tao and Vu in their study of symmetric random matrices. In this
paper, we obtain an essentially optimal bound for this problem, as conjectured
by Nguyen and Vu.
  Specifically, if $Q(\xi_{1},\dots,\xi_{n})$ "robustly depends on at least $m$
of the $\xi_{i}$" in the sense that there is no way to pin down the value of
$Q(\xi_{1},\dots,\xi_{n})$ by fixing values for fewer than $m$ of the variables
$\xi_{i}$, then we have $\Pr[Q(\xi_{1},\dots,\xi_{n})=0]\le O(1/\sqrt{m})$.
This also implies a similar result in the case where $\xi_{1},\dots,\xi_{n}$
have arbitrary distributions.
  Our proof combines a number of ideas that may be of independent interest,
including an inductive decoupling scheme that reduces quadratic
anticoncentration problems to high-dimensional linear anticoncentration
problems. Also, one application of our main result is the resolution of a
conjecture of Alon, Hefetz, Krivelevich and Tyomkyn related to graph
inducibility.
