---
OA_place: repository
OA_type: green
_id: '19496'
abstract:
- lang: eng
  text: We introduce the notions of scale for sets and measures on metric space by
    generalizing the usual notions of dimension. Several versions of scales are introduced
    such as Hausdorff, packing, box, local and quantization. They are defined for
    different growth, allowing a refined study of infinite dimensional spaces. We
    prove general theorems comparing the different versions of scales. They are applied
    to describe geometries of ergodic decompositions, of the Wiener measure and from
    functional spaces. The first application solves a problem of Berger on the notions
    of emergence (2020); the second lies in the geometry of the Wiener measure and
    extends the work of Dereich–Lifshits (2005); the last refines Kolmogorov–Tikhomirov
    (1958) study on finitely differentiable functions.
article_number: '15'
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Mathieu
  full_name: Helfter, Mathieu
  id: 7d296fbe-e2c6-11ee-84d3-d5c2945f9a57
  last_name: Helfter
citation:
  ama: Helfter M. Scales. <i>Mathematische Zeitschrift</i>. 2025;310. doi:<a href="https://doi.org/10.1007/s00209-025-03719-5">10.1007/s00209-025-03719-5</a>
  apa: Helfter, M. (2025). Scales. <i>Mathematische Zeitschrift</i>. Springer Nature.
    <a href="https://doi.org/10.1007/s00209-025-03719-5">https://doi.org/10.1007/s00209-025-03719-5</a>
  chicago: Helfter, Mathieu. “Scales.” <i>Mathematische Zeitschrift</i>. Springer
    Nature, 2025. <a href="https://doi.org/10.1007/s00209-025-03719-5">https://doi.org/10.1007/s00209-025-03719-5</a>.
  ieee: M. Helfter, “Scales,” <i>Mathematische Zeitschrift</i>, vol. 310. Springer
    Nature, 2025.
  ista: Helfter M. 2025. Scales. Mathematische Zeitschrift. 310, 15.
  mla: Helfter, Mathieu. “Scales.” <i>Mathematische Zeitschrift</i>, vol. 310, 15,
    Springer Nature, 2025, doi:<a href="https://doi.org/10.1007/s00209-025-03719-5">10.1007/s00209-025-03719-5</a>.
  short: M. Helfter, Mathematische Zeitschrift 310 (2025).
corr_author: '1'
date_created: 2025-04-06T22:01:32Z
date_published: 2025-05-01T00:00:00Z
date_updated: 2025-09-30T11:31:00Z
day: '01'
department:
- _id: VaKa
doi: 10.1007/s00209-025-03719-5
external_id:
  arxiv:
  - '2206.05231'
  isi:
  - '001450830300001'
intvolume: '       310'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2206.05231
month: '05'
oa: 1
oa_version: Preprint
publication: Mathematische Zeitschrift
publication_identifier:
  eissn:
  - 1432-1823
  issn:
  - 0025-5874
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Scales
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 310
year: '2025'
...
---
OA_place: publisher
OA_type: hybrid
_id: '19776'
abstract:
- lang: eng
  text: We use the circle method to prove that a density 1 of elements in Fq[t] are
    representable as a sum of three cubes of essentially minimal degree from Fq[t],
    assuming the Ratios Conjecture and that char(Fq)>3. Roughly speaking, to do so,
    we upgrade an order of magnitude result to a full asymptotic formula that was
    conjectured by Hooley in the number field setting.
acknowledgement: We thank Alexandra Florea for discussions on cubic Gauss sums over
  function fields, in addition to the anonymous referee for helpful comments. While
  working on this paper the first two authors were supported by a FWF grant (DOI 10.55776/P36278)
  and the third author was supported by the European Union’s Horizon 2020 research
  and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 101034413.
  Open access funding provided by Institute of Science and Technology (IST Austria).
article_number: '65'
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- 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: Jakob
  full_name: Glas, Jakob
  id: d6423cba-dc74-11ea-a0a7-ee61689ff5fb
  last_name: Glas
- first_name: Victor
  full_name: Wang, Victor
  id: 76096395-aea4-11ed-a680-ab8ebbd3f1b9
  last_name: Wang
  orcid: 0000-0002-0704-7026
citation:
  ama: Browning TD, Glas J, Wang V. Optimal sums of three cubes in Fq[t]. <i>Mathematische
    Zeitschrift</i>. 2025;310(4). doi:<a href="https://doi.org/10.1007/s00209-025-03765-z">10.1007/s00209-025-03765-z</a>
  apa: Browning, T. D., Glas, J., &#38; Wang, V. (2025). Optimal sums of three cubes
    in Fq[t]. <i>Mathematische Zeitschrift</i>. Springer Nature. <a href="https://doi.org/10.1007/s00209-025-03765-z">https://doi.org/10.1007/s00209-025-03765-z</a>
  chicago: Browning, Timothy D, Jakob Glas, and Victor Wang. “Optimal Sums of Three
    Cubes in Fq[T].” <i>Mathematische Zeitschrift</i>. Springer Nature, 2025. <a href="https://doi.org/10.1007/s00209-025-03765-z">https://doi.org/10.1007/s00209-025-03765-z</a>.
  ieee: T. D. Browning, J. Glas, and V. Wang, “Optimal sums of three cubes in Fq[t],”
    <i>Mathematische Zeitschrift</i>, vol. 310, no. 4. Springer Nature, 2025.
  ista: Browning TD, Glas J, Wang V. 2025. Optimal sums of three cubes in Fq[t]. Mathematische
    Zeitschrift. 310(4), 65.
  mla: Browning, Timothy D., et al. “Optimal Sums of Three Cubes in Fq[T].” <i>Mathematische
    Zeitschrift</i>, vol. 310, no. 4, 65, Springer Nature, 2025, doi:<a href="https://doi.org/10.1007/s00209-025-03765-z">10.1007/s00209-025-03765-z</a>.
  short: T.D. Browning, J. Glas, V. Wang, Mathematische Zeitschrift 310 (2025).
corr_author: '1'
date_created: 2025-06-03T07:30:21Z
date_published: 2025-05-23T00:00:00Z
date_updated: 2025-09-30T12:43:41Z
day: '23'
ddc:
- '510'
department:
- _id: TiBr
doi: 10.1007/s00209-025-03765-z
ec_funded: 1
external_id:
  arxiv:
  - '2408.03668 '
  isi:
  - '001494367000001'
file:
- access_level: open_access
  checksum: 6f71e25740c28257bf89b8bf116c2b4d
  content_type: application/pdf
  creator: dernst
  date_created: 2025-06-03T08:28:14Z
  date_updated: 2025-06-03T08:28:14Z
  file_id: '19782'
  file_name: 2025_MathZeitschrift_Browning.pdf
  file_size: 461622
  relation: main_file
  success: 1
file_date_updated: 2025-06-03T08:28:14Z
has_accepted_license: '1'
intvolume: '       310'
isi: 1
issue: '4'
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
month: '05'
oa: 1
oa_version: Published Version
project:
- _id: bd8a4fdc-d553-11ed-ba76-80a0167441a3
  grant_number: P36278
  name: Rational curves via function field analytic number theory
- _id: fc2ed2f7-9c52-11eb-aca3-c01059dda49c
  call_identifier: H2020
  grant_number: '101034413'
  name: 'IST-BRIDGE: International postdoctoral program'
publication: Mathematische Zeitschrift
publication_identifier:
  eissn:
  - 1432-1823
  issn:
  - 0025-5874
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Optimal sums of three cubes in Fq[t]
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: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 310
year: '2025'
...
---
_id: '12210'
abstract:
- lang: eng
  text: "The aim of this paper is to find new estimates for the norms of functions
    of a (minus) distinguished Laplace operator L on the ‘ax+b’ groups. The central
    part is devoted to spectrally localized wave propagators, that is, functions of
    the type ψ(L−−√)exp(itL−−√), with ψ∈C0(R). We show that for t→+∞, the convolution
    kernel kt of this operator satisfies\r\n∥kt∥1≍t,∥kt∥∞≍1,\r\nso that the upper
    estimates of D. Müller and C. Thiele (Studia Math., 2007) are sharp. As a necessary
    component, we recall the Plancherel density of L and spend certain time presenting
    and comparing different approaches to its calculation. Using its explicit form,
    we estimate uniform norms of several functions of the shifted Laplace-Beltrami
    operator Δ~, closely related to L. The functions include in particular exp(−tΔ~γ),
    t>0,γ>0, and (Δ~−z)s, with complex z, s."
acknowledgement: "Yu. K. thanks Professor Waldemar Hebisch for valuable discussions
  on the general context of multipliers on Lie groups. This work was started during
  an ICL-CNRS fellowship of the second\r\nnamed author at the Imperial College London.
  Yu. K. is supported by the ANR-19-CE40-0002 grant of the French National Research
  Agency (ANR). H. Z. is supported by the European Union’s Horizon 2020 research and
  innovation programme under the Marie Skłodowska-Curie grant agreement No. 754411
  and the Lise Meitner fellowship, Austrian Science Fund (FWF) M3337. R. A. was supported
  by the EPSRC grant EP/R003025. M. R. is supported by the EPSRC grant EP/R003025/2
  and by the FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial Differential Equations."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Rauan
  full_name: Akylzhanov, Rauan
  last_name: Akylzhanov
- first_name: Yulia
  full_name: Kuznetsova, Yulia
  last_name: Kuznetsova
- first_name: Michael
  full_name: Ruzhansky, Michael
  last_name: Ruzhansky
- first_name: Haonan
  full_name: Zhang, Haonan
  id: D8F41E38-9E66-11E9-A9E2-65C2E5697425
  last_name: Zhang
citation:
  ama: Akylzhanov R, Kuznetsova Y, Ruzhansky M, Zhang H. Norms of certain functions
    of a distinguished Laplacian on the ax + b groups. <i>Mathematische Zeitschrift</i>.
    2022;302(4):2327-2352. doi:<a href="https://doi.org/10.1007/s00209-022-03143-z">10.1007/s00209-022-03143-z</a>
  apa: Akylzhanov, R., Kuznetsova, Y., Ruzhansky, M., &#38; Zhang, H. (2022). Norms
    of certain functions of a distinguished Laplacian on the ax + b groups. <i>Mathematische
    Zeitschrift</i>. Springer Nature. <a href="https://doi.org/10.1007/s00209-022-03143-z">https://doi.org/10.1007/s00209-022-03143-z</a>
  chicago: Akylzhanov, Rauan, Yulia Kuznetsova, Michael Ruzhansky, and Haonan Zhang.
    “Norms of Certain Functions of a Distinguished Laplacian on the Ax + b Groups.”
    <i>Mathematische Zeitschrift</i>. Springer Nature, 2022. <a href="https://doi.org/10.1007/s00209-022-03143-z">https://doi.org/10.1007/s00209-022-03143-z</a>.
  ieee: R. Akylzhanov, Y. Kuznetsova, M. Ruzhansky, and H. Zhang, “Norms of certain
    functions of a distinguished Laplacian on the ax + b groups,” <i>Mathematische
    Zeitschrift</i>, vol. 302, no. 4. Springer Nature, pp. 2327–2352, 2022.
  ista: Akylzhanov R, Kuznetsova Y, Ruzhansky M, Zhang H. 2022. Norms of certain functions
    of a distinguished Laplacian on the ax + b groups. Mathematische Zeitschrift.
    302(4), 2327–2352.
  mla: Akylzhanov, Rauan, et al. “Norms of Certain Functions of a Distinguished Laplacian
    on the Ax + b Groups.” <i>Mathematische Zeitschrift</i>, vol. 302, no. 4, Springer
    Nature, 2022, pp. 2327–52, doi:<a href="https://doi.org/10.1007/s00209-022-03143-z">10.1007/s00209-022-03143-z</a>.
  short: R. Akylzhanov, Y. Kuznetsova, M. Ruzhansky, H. Zhang, Mathematische Zeitschrift
    302 (2022) 2327–2352.
date_created: 2023-01-16T09:45:31Z
date_published: 2022-12-01T00:00:00Z
date_updated: 2025-04-14T07:44:00Z
day: '01'
department:
- _id: JaMa
doi: 10.1007/s00209-022-03143-z
ec_funded: 1
external_id:
  arxiv:
  - '2101.00584'
  isi:
  - '000859680700001'
intvolume: '       302'
isi: 1
issue: '4'
keyword:
- General Mathematics
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/2101.00584
month: '12'
oa: 1
oa_version: Preprint
page: 2327-2352
project:
- _id: 260C2330-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '754411'
  name: ISTplus - Postdoctoral Fellowships
- _id: eb958bca-77a9-11ec-83b8-c565cb50d8d6
  grant_number: M03337
  name: Curvature-dimension in noncommutative analysis
publication: Mathematische Zeitschrift
publication_identifier:
  eissn:
  - 1432-1823
  issn:
  - 0025-5874
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Norms of certain functions of a distinguished Laplacian on the ax + b groups
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 302
year: '2022'
...
---
OA_place: repository
OA_type: green
_id: '19492'
abstract:
- lang: eng
  text: Kuroda’s formula relates the class number of a multiquadratic number field
    K to the class numbers of its quadratic subfields ki. A key component in this
    formula is the unit group index (math formular). We study how Q(K) behaves on
    average in certain natural families of totally real biquadratic fields K parametrized
    by prime numbers.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Yik Tung
  full_name: Chan, Yik Tung
  id: c4c0afc8-9262-11ed-9231-d8b0bc743af1
  last_name: Chan
  orcid: 0000-0001-8467-4106
- first_name: Djordjo
  full_name: Milovic, Djordjo
  last_name: Milovic
citation:
  ama: Chan S, Milovic D. Kuroda’s formula and arithmetic statistics. <i>Mathematische
    Zeitschrift</i>. 2021;300(2):1509-1527. doi:<a href="https://doi.org/10.1007/s00209-021-02823-6">10.1007/s00209-021-02823-6</a>
  apa: Chan, S., &#38; Milovic, D. (2021). Kuroda’s formula and arithmetic statistics.
    <i>Mathematische Zeitschrift</i>. Springer Nature. <a href="https://doi.org/10.1007/s00209-021-02823-6">https://doi.org/10.1007/s00209-021-02823-6</a>
  chicago: Chan, Stephanie, and Djordjo Milovic. “Kuroda’s Formula and Arithmetic
    Statistics.” <i>Mathematische Zeitschrift</i>. Springer Nature, 2021. <a href="https://doi.org/10.1007/s00209-021-02823-6">https://doi.org/10.1007/s00209-021-02823-6</a>.
  ieee: S. Chan and D. Milovic, “Kuroda’s formula and arithmetic statistics,” <i>Mathematische
    Zeitschrift</i>, vol. 300, no. 2. Springer Nature, pp. 1509–1527, 2021.
  ista: Chan S, Milovic D. 2021. Kuroda’s formula and arithmetic statistics. Mathematische
    Zeitschrift. 300(2), 1509–1527.
  mla: Chan, Stephanie, and Djordjo Milovic. “Kuroda’s Formula and Arithmetic Statistics.”
    <i>Mathematische Zeitschrift</i>, vol. 300, no. 2, Springer Nature, 2021, pp.
    1509–27, doi:<a href="https://doi.org/10.1007/s00209-021-02823-6">10.1007/s00209-021-02823-6</a>.
  short: S. Chan, D. Milovic, Mathematische Zeitschrift 300 (2021) 1509–1527.
date_created: 2025-04-05T10:51:04Z
date_published: 2021-08-17T00:00:00Z
date_updated: 2025-07-10T11:51:48Z
day: '17'
doi: 10.1007/s00209-021-02823-6
extern: '1'
external_id:
  arxiv:
  - '1905.09745'
intvolume: '       300'
issue: '2'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.1905.09745
month: '08'
oa: 1
oa_version: Preprint
page: 1509-1527
publication: Mathematische Zeitschrift
publication_identifier:
  eissn:
  - 1432-1823
  issn:
  - 0025-5874
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Kuroda’s formula and arithmetic statistics
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 300
year: '2021'
...
---
_id: '9260'
abstract:
- lang: eng
  text: We study the density of rational points on a higher-dimensional orbifold (Pn−1,Δ)
    when Δ is a Q-divisor involving hyperplanes. This allows us to address a question
    of Tanimoto about whether the set of rational points on such an orbifold constitutes
    a thin set. Our approach relies on the Hardy–Littlewood circle method to first
    study an asymptotic version of Waring’s problem for mixed powers. In doing so
    we make crucial use of the recent resolution of the main conjecture in Vinogradov’s
    mean value theorem, due to Bourgain–Demeter–Guth and Wooley.
acknowledgement: While working on this paper the authors were both supported by EPSRC
  grant EP/P026710/1, and the second author received additional support from the NWO
  Veni Grant 016.Veni.192.047. Thanks are due to Marta Pieropan, Arne Smeets and Sho
  Tanimoto for useful conversations related to this topic, and to the anonymous referee
  for numerous helpful suggestions.
article_processing_charge: No
article_type: original
author:
- 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: Shuntaro
  full_name: Yamagishi, Shuntaro
  last_name: Yamagishi
citation:
  ama: Browning TD, Yamagishi S. Arithmetic of higher-dimensional orbifolds and a
    mixed Waring problem. <i>Mathematische Zeitschrift</i>. 2021;299:1071–1101. doi:<a
    href="https://doi.org/10.1007/s00209-021-02695-w">10.1007/s00209-021-02695-w</a>
  apa: Browning, T. D., &#38; Yamagishi, S. (2021). Arithmetic of higher-dimensional
    orbifolds and a mixed Waring problem. <i>Mathematische Zeitschrift</i>. Springer
    Nature. <a href="https://doi.org/10.1007/s00209-021-02695-w">https://doi.org/10.1007/s00209-021-02695-w</a>
  chicago: Browning, Timothy D, and Shuntaro Yamagishi. “Arithmetic of Higher-Dimensional
    Orbifolds and a Mixed Waring Problem.” <i>Mathematische Zeitschrift</i>. Springer
    Nature, 2021. <a href="https://doi.org/10.1007/s00209-021-02695-w">https://doi.org/10.1007/s00209-021-02695-w</a>.
  ieee: T. D. Browning and S. Yamagishi, “Arithmetic of higher-dimensional orbifolds
    and a mixed Waring problem,” <i>Mathematische Zeitschrift</i>, vol. 299. Springer
    Nature, pp. 1071–1101, 2021.
  ista: Browning TD, Yamagishi S. 2021. Arithmetic of higher-dimensional orbifolds
    and a mixed Waring problem. Mathematische Zeitschrift. 299, 1071–1101.
  mla: Browning, Timothy D., and Shuntaro Yamagishi. “Arithmetic of Higher-Dimensional
    Orbifolds and a Mixed Waring Problem.” <i>Mathematische Zeitschrift</i>, vol.
    299, Springer Nature, 2021, pp. 1071–1101, doi:<a href="https://doi.org/10.1007/s00209-021-02695-w">10.1007/s00209-021-02695-w</a>.
  short: T.D. Browning, S. Yamagishi, Mathematische Zeitschrift 299 (2021) 1071–1101.
date_created: 2021-03-21T23:01:21Z
date_published: 2021-03-05T00:00:00Z
date_updated: 2025-04-14T09:25:44Z
day: '05'
ddc:
- '510'
department:
- _id: TiBr
doi: 10.1007/s00209-021-02695-w
external_id:
  isi:
  - '000625573800002'
file:
- access_level: open_access
  checksum: 8ed9f49568806894744096dbbca0ad7b
  content_type: application/pdf
  creator: dernst
  date_created: 2021-03-22T12:41:26Z
  date_updated: 2021-03-22T12:41:26Z
  file_id: '9279'
  file_name: 2021_MathZeitschrift_Browning.pdf
  file_size: 492685
  relation: main_file
  success: 1
file_date_updated: 2021-03-22T12:41:26Z
has_accepted_license: '1'
intvolume: '       299'
isi: 1
language:
- iso: eng
month: '03'
oa: 1
oa_version: Published Version
page: 1071–1101
project:
- _id: 26A8D266-B435-11E9-9278-68D0E5697425
  grant_number: EP-P026710-2
  name: Between rational and integral points
publication: Mathematische Zeitschrift
publication_identifier:
  eissn:
  - 1432-1823
  issn:
  - 0025-5874
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Arithmetic of higher-dimensional orbifolds and a mixed Waring 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: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 299
year: '2021'
...