Resolution of the quadratic Littlewood–Offord problem Kwan, Matthew Alan ; https://orcid.org/0000-0002-4003-7567 Sauermann, Lisa ddc:510 Consider a quadratic polynomial Q(ξ1, . . . , ξn) of independent Rademacher random variables ξ1, . . . , ξn. To what extent can Q(ξ1, . . . , ξn) 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(ξ1, . . . , ξn) ‘robustly depends on at least m of the ξi’ in the sense that there is no way to pin down the value of Q(ξ1, . . . , ξn) by fixing values for fewer than m of the variables ξi, then we have Pr[Q(ξ1, . . . , ξn) = 0] ≤ O(1/√m). This also implies a similar result in the case where ξ1, . . . , ξn 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. Cambridge University Press 2025 info:eu-repo/semantics/article doc-type:article text http://purl.org/coar/resource_type/c_2df8fbb1 https://research-explorer.ista.ac.at/record/21706 https://research-explorer.ista.ac.at/download/21706/21787 Kwan MA, Sauermann L. Resolution of the quadratic Littlewood–Offord problem. <i>Compositio Mathematica</i>. 2025;161(12):3089-3139. doi:<a href="https://doi.org/10.1112/S0010437X25102789">10.1112/S0010437X25102789</a> eng info:eu-repo/semantics/altIdentifier/doi/10.1112/S0010437X25102789 info:eu-repo/semantics/altIdentifier/issn/0010-437X info:eu-repo/semantics/altIdentifier/e-issn/1570-5846 info:eu-repo/semantics/altIdentifier/arxiv/2312.13826 https://creativecommons.org/licenses/by/4.0/ info:eu-repo/semantics/openAccess