article published yes Matthew Alan Kwan author 5fca0887-a1db-11eb-95d1-ca9d5e0453b30000-0002-4003-7567 Lisa Sauermann author MaKw department Randomness and structure in combinatorics project 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. https://research-explorer.ista.ac.at/download/21706/21787/2025_CompositioMath_Kwan.pdf application/pdfno https://creativecommons.org/licenses/by/4.0/ Cambridge University Press2025 eng 0010-437X 1570-5846 2312.1382610.1112/S0010437X25102789 161123089-3139 M. A. Kwan and L. Sauermann, “Resolution of the quadratic Littlewood–Offord problem,” <i>Compositio Mathematica</i>, vol. 161, no. 12. Cambridge University Press, pp. 3089–3139, 2025. M.A. Kwan, L. Sauermann, Compositio Mathematica 161 (2025) 3089–3139. Kwan, Matthew Alan, and Lisa Sauermann. “Resolution of the Quadratic Littlewood–Offord Problem.” <i>Compositio Mathematica</i>, vol. 161, no. 12, Cambridge University Press, 2025, pp. 3089–139, doi:<a href="https://doi.org/10.1112/S0010437X25102789">10.1112/S0010437X25102789</a>. 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> Kwan, M. A., &#38; Sauermann, L. (2025). Resolution of the quadratic Littlewood–Offord problem. <i>Compositio Mathematica</i>. Cambridge University Press. <a href="https://doi.org/10.1112/S0010437X25102789">https://doi.org/10.1112/S0010437X25102789</a> Kwan MA, Sauermann L. 2025. Resolution of the quadratic Littlewood–Offord problem. Compositio Mathematica. 161(12), 3089–3139. Kwan, Matthew Alan, and Lisa Sauermann. “Resolution of the Quadratic Littlewood–Offord Problem.” <i>Compositio Mathematica</i>. Cambridge University Press, 2025. <a href="https://doi.org/10.1112/S0010437X25102789">https://doi.org/10.1112/S0010437X25102789</a>. 217062026-04-12T22:01:48Z2026-05-04T09:42:57Z