[{"arxiv":1,"article_processing_charge":"Yes (via OA deal)","page":"3089-3139","external_id":{"arxiv":["2312.13826"]},"project":[{"name":"Randomness and structure in combinatorics","_id":"bd95085b-d553-11ed-ba76-e55d3349be45","grant_number":"101076777"}],"publication":"Compositio Mathematica","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. "}],"volume":161,"file_date_updated":"2026-05-04T09:41:25Z","title":"Resolution of the quadratic Littlewood–Offord problem","OA_place":"publisher","issue":"12","quality_controlled":"1","language":[{"iso":"eng"}],"oa":1,"oa_version":"Published Version","department":[{"_id":"MaKw"}],"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.","doi":"10.1112/S0010437X25102789","date_created":"2026-04-12T22:01:48Z","publication_identifier":{"issn":["0010-437X"],"eissn":["1570-5846"]},"_id":"21706","month":"12","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","type":"journal_article","scopus_import":"1","citation":{"ista":"Kwan MA, Sauermann L. 2025. Resolution of the quadratic Littlewood–Offord problem. Compositio Mathematica. 161(12), 3089–3139.","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>","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>.","short":"M.A. Kwan, L. Sauermann, Compositio Mathematica 161 (2025) 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>"},"intvolume":"       161","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"has_accepted_license":"1","OA_type":"hybrid","corr_author":"1","ddc":["510"],"date_published":"2025-12-01T00:00:00Z","author":[{"first_name":"Matthew Alan","last_name":"Kwan","orcid":"0000-0002-4003-7567","full_name":"Kwan, Matthew Alan","id":"5fca0887-a1db-11eb-95d1-ca9d5e0453b3"},{"last_name":"Sauermann","first_name":"Lisa","full_name":"Sauermann, Lisa"}],"year":"2025","day":"01","PlanS_conform":"1","file":[{"file_name":"2025_CompositioMath_Kwan.pdf","file_id":"21787","success":1,"date_created":"2026-05-04T09:41:25Z","relation":"main_file","checksum":"bd3415bb435da9d0b39f6f9a18c61abb","access_level":"open_access","creator":"dernst","content_type":"application/pdf","date_updated":"2026-05-04T09:41:25Z","file_size":858727}],"license":"https://creativecommons.org/licenses/by/4.0/","status":"public","publication_status":"published","date_updated":"2026-05-04T09:42:57Z","publisher":"Cambridge University Press","article_type":"original"},{"external_id":{"isi":["000667289300001"],"arxiv":["1909.03266"]},"article_processing_charge":"No","page":"1610-1651","arxiv":1,"title":"The distribution of the maximum of partial sums of Kloosterman sums and other trace functions","abstract":[{"lang":"eng","text":"In this paper, we investigate the distribution of the maximum of partial sums of families of  m -periodic complex-valued functions satisfying certain conditions. We obtain precise uniform estimates for the distribution function of this maximum in a near-optimal range. Our results apply to partial sums of Kloosterman sums and other families of  ℓ -adic trace functions, and are as strong as those obtained by Bober, Goldmakher, Granville and Koukoulopoulos for character sums. In particular, we improve on the recent work of the third author for Birch sums. However, unlike character sums, we are able to construct families of  m -periodic complex-valued functions which satisfy our conditions, but for which the Pólya–Vinogradov inequality is sharp."}],"publication":"Compositio Mathematica","volume":157,"department":[{"_id":"TiBr"}],"oa_version":"Preprint","oa":1,"issue":"7","quality_controlled":"1","language":[{"iso":"eng"}],"scopus_import":"1","citation":{"chicago":"Autissier, Pascal, Dante Bonolis, and Youness Lamzouri. “The Distribution of the Maximum of Partial Sums of Kloosterman Sums and Other Trace Functions.” <i>Compositio Mathematica</i>. Cambridge University Press, 2021. <a href=\"https://doi.org/10.1112/s0010437x21007351\">https://doi.org/10.1112/s0010437x21007351</a>.","ama":"Autissier P, Bonolis D, Lamzouri Y. The distribution of the maximum of partial sums of Kloosterman sums and other trace functions. <i>Compositio Mathematica</i>. 2021;157(7):1610-1651. doi:<a href=\"https://doi.org/10.1112/s0010437x21007351\">10.1112/s0010437x21007351</a>","ieee":"P. Autissier, D. Bonolis, and Y. Lamzouri, “The distribution of the maximum of partial sums of Kloosterman sums and other trace functions,” <i>Compositio Mathematica</i>, vol. 157, no. 7. Cambridge University Press, pp. 1610–1651, 2021.","mla":"Autissier, Pascal, et al. “The Distribution of the Maximum of Partial Sums of Kloosterman Sums and Other Trace Functions.” <i>Compositio Mathematica</i>, vol. 157, no. 7, Cambridge University Press, 2021, pp. 1610–51, doi:<a href=\"https://doi.org/10.1112/s0010437x21007351\">10.1112/s0010437x21007351</a>.","short":"P. Autissier, D. Bonolis, Y. Lamzouri, Compositio Mathematica 157 (2021) 1610–1651.","apa":"Autissier, P., Bonolis, D., &#38; Lamzouri, Y. (2021). The distribution of the maximum of partial sums of Kloosterman sums and other trace functions. <i>Compositio Mathematica</i>. Cambridge University Press. <a href=\"https://doi.org/10.1112/s0010437x21007351\">https://doi.org/10.1112/s0010437x21007351</a>","ista":"Autissier P, Bonolis D, Lamzouri Y. 2021. The distribution of the maximum of partial sums of Kloosterman sums and other trace functions. Compositio Mathematica. 157(7), 1610–1651."},"month":"06","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","type":"journal_article","date_created":"2022-02-01T08:10:43Z","doi":"10.1112/s0010437x21007351","_id":"10711","publication_identifier":{"eissn":["1570-5846"],"issn":["0010-437X"]},"acknowledgement":"We would like to thank the anonymous referees for carefully reading the paper and for their remarks and suggestions.","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1909.03266"}],"keyword":["Algebra and Number Theory"],"isi":1,"intvolume":"       157","author":[{"full_name":"Autissier, Pascal","last_name":"Autissier","first_name":"Pascal"},{"last_name":"Bonolis","first_name":"Dante","id":"6A459894-5FDD-11E9-AF35-BB24E6697425","full_name":"Bonolis, Dante"},{"last_name":"Lamzouri","first_name":"Youness","full_name":"Lamzouri, Youness"}],"date_published":"2021-06-28T00:00:00Z","corr_author":"1","day":"28","year":"2021","publisher":"Cambridge University Press","article_type":"original","date_updated":"2024-10-21T06:02:06Z","status":"public","publication_status":"published"},{"oa":1,"oa_version":"Preprint","OA_place":"repository","language":[{"iso":"eng"}],"issue":"7","type":"journal_article","month":"07","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","citation":{"chicago":"Bright, Maritn, Timothy D Browning, and Daniel Loughran. “Failures of Weak Approximation in Families.” <i>Compositio Mathematica</i>. Cambridge University Press, 2016. <a href=\"https://doi.org/10.1112/S0010437X16007405\">https://doi.org/10.1112/S0010437X16007405</a>.","ama":"Bright M, Browning TD, Loughran D. Failures of weak approximation in families. <i>Compositio Mathematica</i>. 2016;152(7):1435-1475. doi:<a href=\"https://doi.org/10.1112/S0010437X16007405\">10.1112/S0010437X16007405</a>","ieee":"M. Bright, T. D. Browning, and D. Loughran, “Failures of weak approximation in families,” <i>Compositio Mathematica</i>, vol. 152, no. 7. Cambridge University Press, pp. 1435–1475, 2016.","mla":"Bright, Maritn, et al. “Failures of Weak Approximation in Families.” <i>Compositio Mathematica</i>, vol. 152, no. 7, Cambridge University Press, 2016, pp. 1435–75, doi:<a href=\"https://doi.org/10.1112/S0010437X16007405\">10.1112/S0010437X16007405</a>.","short":"M. Bright, T.D. Browning, D. Loughran, Compositio Mathematica 152 (2016) 1435–1475.","apa":"Bright, M., Browning, T. D., &#38; Loughran, D. (2016). Failures of weak approximation in families. <i>Compositio Mathematica</i>. Cambridge University Press. <a href=\"https://doi.org/10.1112/S0010437X16007405\">https://doi.org/10.1112/S0010437X16007405</a>","ista":"Bright M, Browning TD, Loughran D. 2016. Failures of weak approximation in families. Compositio Mathematica. 152(7), 1435–1475."},"main_file_link":[{"open_access":"1","url":" https://doi.org/10.48550/arXiv.1506.01817"}],"acknowledgement":"While working on this paper the second author was supported by ERC grant 306457.","publication_identifier":{"eissn":["1570-5846"],"issn":["0010-437X"]},"_id":"264","date_created":"2018-12-11T11:45:30Z","doi":"10.1112/S0010437X16007405","page":"1435 - 1475","external_id":{"arxiv":["1506.01817"]},"article_processing_charge":"No","arxiv":1,"title":"Failures of weak approximation in families","volume":152,"abstract":[{"text":"Given a family of varieties over a number field, we determine conditions under which there is a Brauer-Manin obstruction to weak approximation for 100% of the fibres which are everywhere locally soluble.","lang":"eng"}],"publication":"Compositio Mathematica","year":"2016","day":"01","date_updated":"2026-05-13T14:57:20Z","publisher":"Cambridge University Press","article_type":"original","publication_status":"published","status":"public","intvolume":"       152","extern":"1","publist_id":"7638","author":[{"last_name":"Bright","first_name":"Maritn","full_name":"Bright, Maritn"},{"first_name":"Timothy D","last_name":"Browning","full_name":"Browning, Timothy D","id":"35827D50-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-8314-0177"},{"full_name":"Loughran, Daniel","last_name":"Loughran","first_name":"Daniel"}],"OA_type":"green","date_published":"2016-07-01T00:00:00Z"}]