---
OA_place: publisher
OA_type: hybrid
PlanS_conform: '1'
_id: '21706'
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. "
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.
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Matthew Alan
  full_name: Kwan, Matthew Alan
  id: 5fca0887-a1db-11eb-95d1-ca9d5e0453b3
  last_name: Kwan
  orcid: 0000-0002-4003-7567
- first_name: Lisa
  full_name: Sauermann, Lisa
  last_name: Sauermann
citation:
  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>
  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>.
  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.
  ista: Kwan MA, Sauermann L. 2025. Resolution of the quadratic Littlewood–Offord
    problem. Compositio Mathematica. 161(12), 3089–3139.
  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.
corr_author: '1'
date_created: 2026-04-12T22:01:48Z
date_published: 2025-12-01T00:00:00Z
date_updated: 2026-05-04T09:42:57Z
day: '01'
ddc:
- '510'
department:
- _id: MaKw
doi: 10.1112/S0010437X25102789
external_id:
  arxiv:
  - '2312.13826'
file:
- access_level: open_access
  checksum: bd3415bb435da9d0b39f6f9a18c61abb
  content_type: application/pdf
  creator: dernst
  date_created: 2026-05-04T09:41:25Z
  date_updated: 2026-05-04T09:41:25Z
  file_id: '21787'
  file_name: 2025_CompositioMath_Kwan.pdf
  file_size: 858727
  relation: main_file
  success: 1
file_date_updated: 2026-05-04T09:41:25Z
has_accepted_license: '1'
intvolume: '       161'
issue: '12'
language:
- iso: eng
month: '12'
oa: 1
oa_version: Published Version
page: 3089-3139
project:
- _id: bd95085b-d553-11ed-ba76-e55d3349be45
  grant_number: '101076777'
  name: Randomness and structure in combinatorics
publication: Compositio Mathematica
publication_identifier:
  eissn:
  - 1570-5846
  issn:
  - 0010-437X
publication_status: published
publisher: Cambridge University Press
quality_controlled: '1'
scopus_import: '1'
status: public
title: Resolution of the quadratic Littlewood–Offord problem
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 161
year: '2025'
...