[{"date_created":"2026-04-12T22:01:48Z","author":[{"orcid":"0000-0002-4003-7567","full_name":"Kwan, Matthew Alan","id":"5fca0887-a1db-11eb-95d1-ca9d5e0453b3","first_name":"Matthew Alan","last_name":"Kwan"},{"first_name":"Lisa","last_name":"Sauermann","full_name":"Sauermann, Lisa"}],"project":[{"name":"Randomness and structure in combinatorics","grant_number":"101076777","_id":"bd95085b-d553-11ed-ba76-e55d3349be45"}],"date_updated":"2026-05-04T09:42:57Z","publication_identifier":{"issn":["0010-437X"],"eissn":["1570-5846"]},"oa":1,"volume":161,"OA_type":"hybrid","external_id":{"arxiv":["2312.13826"]},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa_version":"Published Version","arxiv":1,"abstract":[{"lang":"eng","text":"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\r\nto 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. "}],"status":"public","PlanS_conform":"1","article_processing_charge":"Yes (via OA deal)","article_type":"original","language":[{"iso":"eng"}],"_id":"21706","year":"2025","intvolume":"       161","scopus_import":"1","month":"12","title":"Resolution of the quadratic Littlewood–Offord problem","date_published":"2025-12-01T00:00:00Z","quality_controlled":"1","publication_status":"published","type":"journal_article","file":[{"creator":"dernst","access_level":"open_access","checksum":"bd3415bb435da9d0b39f6f9a18c61abb","file_name":"2025_CompositioMath_Kwan.pdf","date_updated":"2026-05-04T09:41:25Z","date_created":"2026-05-04T09:41:25Z","relation":"main_file","content_type":"application/pdf","file_size":858727,"file_id":"21787","success":1}],"corr_author":"1","department":[{"_id":"MaKw"}],"OA_place":"publisher","has_accepted_license":"1","publisher":"Cambridge University Press","publication":"Compositio Mathematica","ddc":["510"],"page":"3089-3139","file_date_updated":"2026-05-04T09:41:25Z","day":"01","citation":{"short":"M.A. Kwan, L. Sauermann, Compositio Mathematica 161 (2025) 3089–3139.","ieee":"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.","mla":"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>.","ista":"Kwan MA, Sauermann L. 2025. Resolution of the quadratic Littlewood–Offord problem. Compositio Mathematica. 161(12), 3089–3139.","chicago":"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>.","ama":"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>","apa":"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>"},"issue":"12","doi":"10.1112/S0010437X25102789","tmp":{"image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"acknowledgement":"We would like to thank the anonymous referee for a number of helpful comments and suggestions. Matthew Kwan was supported by ERC Starting Grant “RANDSTRUCT” No. 101076777. Lisa Sauermann was supported in part by NSF Award DMS-2100157 and a Sloan Research Fellowship, and in part by the DFG Heisenberg Program.","license":"https://creativecommons.org/licenses/by/4.0/"}]