---
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'
...
---
_id: '10711'
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.
acknowledgement: We would like to thank the anonymous referees for carefully reading
  the paper and for their remarks and suggestions.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Pascal
  full_name: Autissier, Pascal
  last_name: Autissier
- first_name: Dante
  full_name: Bonolis, Dante
  id: 6A459894-5FDD-11E9-AF35-BB24E6697425
  last_name: Bonolis
- first_name: Youness
  full_name: Lamzouri, Youness
  last_name: Lamzouri
citation:
  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>
  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>
  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>.
  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.
  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.
  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.
corr_author: '1'
date_created: 2022-02-01T08:10:43Z
date_published: 2021-06-28T00:00:00Z
date_updated: 2024-10-21T06:02:06Z
day: '28'
department:
- _id: TiBr
doi: 10.1112/s0010437x21007351
external_id:
  arxiv:
  - '1909.03266'
  isi:
  - '000667289300001'
intvolume: '       157'
isi: 1
issue: '7'
keyword:
- Algebra and Number Theory
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1909.03266
month: '06'
oa: 1
oa_version: Preprint
page: 1610-1651
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: The distribution of the maximum of partial sums of Kloosterman sums and other
  trace functions
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 157
year: '2021'
...
---
OA_place: repository
OA_type: green
_id: '264'
abstract:
- lang: eng
  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.
acknowledgement: While working on this paper the second author was supported by ERC
  grant 306457.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Maritn
  full_name: Bright, Maritn
  last_name: Bright
- first_name: Timothy D
  full_name: Browning, Timothy D
  id: 35827D50-F248-11E8-B48F-1D18A9856A87
  last_name: Browning
  orcid: 0000-0002-8314-0177
- first_name: Daniel
  full_name: Loughran, Daniel
  last_name: Loughran
citation:
  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>
  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>
  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>.
  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.
  ista: Bright M, Browning TD, Loughran D. 2016. Failures of weak approximation in
    families. Compositio Mathematica. 152(7), 1435–1475.
  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.
date_created: 2018-12-11T11:45:30Z
date_published: 2016-07-01T00:00:00Z
date_updated: 2026-05-13T14:57:20Z
day: '01'
doi: 10.1112/S0010437X16007405
extern: '1'
external_id:
  arxiv:
  - '1506.01817'
intvolume: '       152'
issue: '7'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: ' https://doi.org/10.48550/arXiv.1506.01817'
month: '07'
oa: 1
oa_version: Preprint
page: 1435 - 1475
publication: Compositio Mathematica
publication_identifier:
  eissn:
  - 1570-5846
  issn:
  - 0010-437X
publication_status: published
publisher: Cambridge University Press
publist_id: '7638'
status: public
title: Failures of weak approximation in families
type: journal_article
user_id: ba8df636-2132-11f1-aed0-ed93e2281fdd
volume: 152
year: '2016'
...