On the resilience of the quadratic Littlewood-Offord problem
<|reference_start|>On the resilience of the quadratic Littlewood-Offord problem: 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.<|reference_end|>
@article{aigner-horev2024on,
    title={On the resilience of the quadratic Littlewood-Offord problem},
    author={Elad Aigner-Horev, and Daniel Rosenberg, and Roi Weiss},
    journal={arXiv preprint arXiv:2402.10504},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.10504},
    primaryClass={math.PR cs.IT cs.LG math.CO math.IT stat.ML}
}