  We study the statistical resilience of the anti-concentration properties of
Rademacher polynomials in face of adversarial deterministic noise taking the
form of sign-flips. Given a multilinear polynomial $f:\mathbb{R}^n \to
\mathbb{R}$ and a Rademacher vector $\boldsymbol{\xi} \in \{\pm 1\}^n$ (with
independent entries), our results provide probabilistic lower bound estimations
on the number of sign-flips that $\boldsymbol{\xi}$ can sustain without
``inflating" the atom probability $\sup_{x \in \mathbb{R} }
\mathbb{P}\{f(\boldsymbol{\xi}) = x\}$ otherwise resulting in an adversarially
biased distribution. Special emphasis is put on bilinear and quadratic forms,
for which strengthened estimates are attained. From a computational
perspective, our results in this venue are instance-bound in such a way that
allows for an efficient computation of the statistical resilience guarantees
from the quadratic polynomial itself directly. All of our probabilistic lower
bound resilience guarantees are asymptotically tight.
  On route, we provide a short proof for a new small-ball probability estimate
fitting Rademacher multilinear polynomials $f: \mathbb{R}^n \to \mathbb{R}$
removeing a polylog-factor from the classical Meka-Nguyen-Vu bound provided the
coefficients are independent of $n$ (dimension-free, hereafter). This removal
was conjectured to be possible by Meka-Nguyen-Vu regardless of our assumption.
Bilinear Rademacher forms with dimension-free coefficients arise naturally in
Combinatorics and specifically in the dense case of the edge-statistics
conjecture posed by Alon, Hefetz, Krivelevich, and Tyomkyn. This case of the
conjecture was resolved by Kwan and Sauermann. Replacing the appeal to the
Meka-Nguyen-Vu classical bound in the work of Kwan, Sudakov, and Tran with our
shortly proved result attains an additional proof of the dense case of the
edge-statistics conjecture.
