---
OA_place: publisher
OA_type: hybrid
PlanS_conform: '1'
_id: '18930'
abstract:
- lang: eng
text: "We study sumsets \U0001D49C + ℬ in the set of squares \U0001D4AE (and, more
generally, in the set of kth powers \U0001D4AEk, where k ≥2 is an integer). It
is known by a result of Gyarmati that \U0001D49C + ℬ ⊂ \U0001D4AEk ∩[1,N] implies
that min(|\U0001D49C|,|ℬ|) =Ok(logN). Here, we study how the upper bound on |ℬ|
decreases, when the size of |\U0001D49C| increases (or vice versa). In particular,
if |\U0001D49C| ≥ Ck1m m(logN)1m , then |ℬ| = Ok(m2logN), for sufficiently large
N, a positive integer m and an explicit constant C > 0. For example, with m ∼
loglogN this gives: If |\U0001D49C| ≥ CkloglogN,then |ℬ| = Ok(logN(loglogN)2)."
acknowledgement: This manuscript grew out of the second author’s MSc Thesis at Graz
University of Technology [34]. C. Elsholtz is supported by a joint FWF-ANR project
ArithRand, grant numbers FWF I 4945-N and ANR-20-CE91-0006. Both authors would like
to thank Igor Shparlinski for drawing our attention to related character sum estimates.
Furthermore, we would like to thank the referee for a careful reading of the paper.
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: Christian
full_name: Elsholtz, Christian
last_name: Elsholtz
- first_name: Lena
full_name: Wurzinger, Lena
id: 50c57d72-32a8-11ee-aeea-d652094d2ccd
last_name: Wurzinger
orcid: 0009-0004-5360-0074
citation:
ama: Elsholtz C, Wurzinger L. Sumsets in the set of squares. The Quarterly Journal
of Mathematics. 2024;75(4):1243-1254. doi:10.1093/qmath/haae044
apa: Elsholtz, C., & Wurzinger, L. (2024). Sumsets in the set of squares. The
Quarterly Journal of Mathematics. Oxford University Press. https://doi.org/10.1093/qmath/haae044
chicago: Elsholtz, Christian, and Lena Wurzinger. “Sumsets in the Set of Squares.”
The Quarterly Journal of Mathematics. Oxford University Press, 2024. https://doi.org/10.1093/qmath/haae044.
ieee: C. Elsholtz and L. Wurzinger, “Sumsets in the set of squares,” The Quarterly
Journal of Mathematics, vol. 75, no. 4. Oxford University Press, pp. 1243–1254,
2024.
ista: Elsholtz C, Wurzinger L. 2024. Sumsets in the set of squares. The Quarterly
Journal of Mathematics. 75(4), 1243–1254.
mla: Elsholtz, Christian, and Lena Wurzinger. “Sumsets in the Set of Squares.” The
Quarterly Journal of Mathematics, vol. 75, no. 4, Oxford University Press,
2024, pp. 1243–54, doi:10.1093/qmath/haae044.
short: C. Elsholtz, L. Wurzinger, The Quarterly Journal of Mathematics 75 (2024)
1243–1254.
corr_author: '1'
date_created: 2025-01-28T06:55:31Z
date_published: 2024-12-01T00:00:00Z
date_updated: 2025-12-04T14:46:28Z
day: '01'
ddc:
- '510'
department:
- _id: TiBr
doi: 10.1093/qmath/haae044
external_id:
isi:
- '001304396600001'
file:
- access_level: open_access
checksum: 1a06e052761d3f1e873463d6f529dd82
content_type: application/pdf
creator: dernst
date_created: 2025-01-28T07:03:51Z
date_updated: 2025-01-28T07:03:51Z
file_id: '18931'
file_name: 2024_QuarterlyJourMath_Elsholtz.pdf
file_size: 424645
relation: main_file
success: 1
file_date_updated: 2025-01-28T07:03:51Z
has_accepted_license: '1'
intvolume: ' 75'
isi: 1
issue: '4'
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
month: '12'
oa: 1
oa_version: Published Version
page: 1243-1254
publication: The Quarterly Journal of Mathematics
publication_identifier:
eissn:
- 1464-3847
issn:
- 0033-5606
publication_status: published
publisher: Oxford University Press
quality_controlled: '1'
scopus_import: '1'
status: public
title: Sumsets in the set of squares
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: 75
year: '2024'
...
---
_id: '17475'
abstract:
- lang: eng
text: "As a discrete analogue of Kac’s celebrated question on ‘hearing the shape
of a drum’ and towards a practical\r\ngraph isomorphism test, it is of interest
to understand which graphs are determined up to isomorphism by\r\ntheir spectrum
(of their adjacency matrix). A striking conjecture in this area, due to van Dam
and Haemers,\r\nis that ‘almost all graphs are determined by their spectrum’,
meaning that the fraction of unlabelled n-vertex\r\ngraphs which are determined
by their spectrum converges to 1 as n → ∞.\r\nIn this paper, we make a step towards
this conjecture, showing that there are exponentially many n-vertex\r\ngraphs
which are determined by their spectrum. This improves on previous bounds (of shape
e\r\nc\r\n√\r\nn\r\n). We also\r\npropose a number of further directions of research.\r\n"
acknowledgement: Matthew Kwan was supported by ERC Starting Grant ‘RANDSTRUCT’ No.
101076777.
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Illya
full_name: Koval, Illya
id: 2eed1f3b-896a-11ed-bdf8-93c7c4bf159e
last_name: Koval
- first_name: Matthew Alan
full_name: Kwan, Matthew Alan
id: 5fca0887-a1db-11eb-95d1-ca9d5e0453b3
last_name: Kwan
orcid: 0000-0002-4003-7567
citation:
ama: Koval I, Kwan MA. Exponentially many graphs are determined by their spectrum.
Quarterly Journal of Mathematics. 2024;75(3):869-899. doi:10.1093/qmath/haae030
apa: Koval, I., & Kwan, M. A. (2024). Exponentially many graphs are determined
by their spectrum. Quarterly Journal of Mathematics. Oxford University
Press. https://doi.org/10.1093/qmath/haae030
chicago: Koval, Illya, and Matthew Alan Kwan. “Exponentially Many Graphs Are Determined
by Their Spectrum.” Quarterly Journal of Mathematics. Oxford University
Press, 2024. https://doi.org/10.1093/qmath/haae030.
ieee: I. Koval and M. A. Kwan, “Exponentially many graphs are determined by their
spectrum,” Quarterly Journal of Mathematics, vol. 75, no. 3. Oxford University
Press, pp. 869–899, 2024.
ista: Koval I, Kwan MA. 2024. Exponentially many graphs are determined by their
spectrum. Quarterly Journal of Mathematics. 75(3), 869–899.
mla: Koval, Illya, and Matthew Alan Kwan. “Exponentially Many Graphs Are Determined
by Their Spectrum.” Quarterly Journal of Mathematics, vol. 75, no. 3, Oxford
University Press, 2024, pp. 869–99, doi:10.1093/qmath/haae030.
short: I. Koval, M.A. Kwan, Quarterly Journal of Mathematics 75 (2024) 869–899.
corr_author: '1'
date_created: 2024-09-01T22:01:07Z
date_published: 2024-06-19T00:00:00Z
date_updated: 2025-09-08T09:09:41Z
day: '19'
ddc:
- '500'
department:
- _id: MaKw
- _id: VaKa
doi: 10.1093/qmath/haae030
external_id:
arxiv:
- '2309.09788'
isi:
- '001249741500001'
file:
- access_level: open_access
checksum: abf200d37ad69e6f2c0750a30296ad97
content_type: application/pdf
creator: cchlebak
date_created: 2024-09-06T12:23:57Z
date_updated: 2024-09-06T12:23:57Z
file_id: '17851'
file_name: 2024_QuJofMath_Koval.pdf
file_size: 946411
relation: main_file
success: 1
file_date_updated: 2024-09-06T12:23:57Z
has_accepted_license: '1'
intvolume: ' 75'
isi: 1
issue: '3'
language:
- iso: eng
month: '06'
oa: 1
oa_version: Published Version
page: 869-899
project:
- _id: bd95085b-d553-11ed-ba76-e55d3349be45
grant_number: '101076777'
name: Randomness and structure in combinatorics
publication: Quarterly Journal of Mathematics
publication_identifier:
eissn:
- 1464-3847
issn:
- 0033-5606
publication_status: published
publisher: Oxford University Press
quality_controlled: '1'
scopus_import: '1'
status: public
title: Exponentially many graphs are determined by their spectrum
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: 75
year: '2024'
...
---
_id: '14717'
abstract:
- lang: eng
text: We count primitive lattices of rank d inside Zn as their covolume tends to
infinity, with respect to certain parameters of such lattices. These parameters
include, for example, the subspace that a lattice spans, namely its projection
to the Grassmannian; its homothety class and its equivalence class modulo rescaling
and rotation, often referred to as a shape. We add to a prior work of Schmidt
by allowing sets in the spaces of parameters that are general enough to conclude
the joint equidistribution of these parameters. In addition to the primitive d-lattices
Λ themselves, we also consider their orthogonal complements in Zn, A1, and show
that the equidistribution occurs jointly for Λ and A1. Finally, our asymptotic
formulas for the number of primitive lattices include an explicit bound on the
error term.
acknowledgement: This work was done when both authors were visiting Institute of Science
and Technology (IST) Austria. T.H. was being supported by Engineering and Physical
Sciences Research Council grant EP/P026710/1. Y.K. had a great time there and is
grateful for the hospitality. The appendix to this paper is largely based on a mini
course T.H. had given at IST in February 2020.
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Tal
full_name: Horesh, Tal
id: C8B7BF48-8D81-11E9-BCA9-F536E6697425
last_name: Horesh
- first_name: Yakov
full_name: Karasik, Yakov
last_name: Karasik
citation:
ama: Horesh T, Karasik Y. Equidistribution of primitive lattices in ℝn. Quarterly
Journal of Mathematics. 2023;74(4):1253-1294. doi:10.1093/qmath/haad008
apa: Horesh, T., & Karasik, Y. (2023). Equidistribution of primitive lattices
in ℝn. Quarterly Journal of Mathematics. Oxford University Press. https://doi.org/10.1093/qmath/haad008
chicago: Horesh, Tal, and Yakov Karasik. “Equidistribution of Primitive Lattices
in ℝn.” Quarterly Journal of Mathematics. Oxford University Press, 2023.
https://doi.org/10.1093/qmath/haad008.
ieee: T. Horesh and Y. Karasik, “Equidistribution of primitive lattices in ℝn,”
Quarterly Journal of Mathematics, vol. 74, no. 4. Oxford University Press,
pp. 1253–1294, 2023.
ista: Horesh T, Karasik Y. 2023. Equidistribution of primitive lattices in ℝn. Quarterly
Journal of Mathematics. 74(4), 1253–1294.
mla: Horesh, Tal, and Yakov Karasik. “Equidistribution of Primitive Lattices in
ℝn.” Quarterly Journal of Mathematics, vol. 74, no. 4, Oxford University
Press, 2023, pp. 1253–94, doi:10.1093/qmath/haad008.
short: T. Horesh, Y. Karasik, Quarterly Journal of Mathematics 74 (2023) 1253–1294.
corr_author: '1'
date_created: 2023-12-31T23:01:03Z
date_published: 2023-12-01T00:00:00Z
date_updated: 2025-09-09T14:03:32Z
day: '01'
ddc:
- '510'
department:
- _id: TiBr
doi: 10.1093/qmath/haad008
external_id:
arxiv:
- '2012.04508'
isi:
- '001005945400001'
file:
- access_level: open_access
checksum: bf29baa9eae8500f3374dbcb80712687
content_type: application/pdf
creator: dernst
date_created: 2024-01-02T07:37:09Z
date_updated: 2024-01-02T07:37:09Z
file_id: '14720'
file_name: 2023_QuarterlyJourMath_Horesh.pdf
file_size: 724748
relation: main_file
success: 1
file_date_updated: 2024-01-02T07:37:09Z
has_accepted_license: '1'
intvolume: ' 74'
isi: 1
issue: '4'
language:
- iso: eng
month: '12'
oa: 1
oa_version: Published Version
page: 1253-1294
project:
- _id: 26A8D266-B435-11E9-9278-68D0E5697425
grant_number: EP-P026710-2
name: Between rational and integral points
publication: Quarterly Journal of Mathematics
publication_identifier:
eissn:
- 1464-3847
issn:
- 0033-5606
publication_status: published
publisher: Oxford University Press
quality_controlled: '1'
scopus_import: '1'
status: public
title: Equidistribution of primitive lattices in ℝn
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: 74
year: '2023'
...