---
OA_type: closed access
_id: '20603'
abstract:
- lang: eng
  text: "We study the growth of sumsets A+B⊂S⊂G, where S does not contain an arithmetic
    progression of length 2k+1, and where G is a commutative group, in which every
    nonzero element has an order of at least 2k+1. More specifically, we show the
    following: if A,B⊂G are sets such that A+B does not contain an arithmetic progression
    of length 2k+1, then\r\n|A+B|≥|A|2k−13k−2|B|k3k−2.\r\nAs an application we derive
    upper bounds on the cardinality of the summands in sumsets A+B+C contained in
    the set of t-th powers, where t≥2 is an integer. In particular, we show that min(|A|,|B|,|C|)≪(logN)4/5
    for t=2, and min(|A|,|B|,|C|)≪t(logN)1/2 for t≥3."
acknowledgement: "The authors would like to thank the referee and Ilya Shkredov for
  comments on the manuscript.\r\nC. E. is supported by a joint FWF-ANR project ArithRand,
  grant numbers FWF I 4945-N and ANR-20-CE91-0006.\r\n"
article_processing_charge: No
article_type: original
author:
- first_name: Christian
  full_name: Elsholtz, Christian
  last_name: Elsholtz
- first_name: Imre Z.
  full_name: Ruzsa, Imre Z.
  last_name: Ruzsa
- 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, Ruzsa IZ, Wurzinger L. Sumset growth in progression-free sets.
    <i>Acta Arithmetica</i>. 2025;220:289-303. doi:<a href="https://doi.org/10.4064/aa250115-14-7">10.4064/aa250115-14-7</a>
  apa: Elsholtz, C., Ruzsa, I. Z., &#38; Wurzinger, L. (2025). Sumset growth in progression-free
    sets. <i>Acta Arithmetica</i>. Institute of Mathematics. <a href="https://doi.org/10.4064/aa250115-14-7">https://doi.org/10.4064/aa250115-14-7</a>
  chicago: Elsholtz, Christian, Imre Z. Ruzsa, and Lena Wurzinger. “Sumset Growth
    in Progression-Free Sets.” <i>Acta Arithmetica</i>. Institute of Mathematics,
    2025. <a href="https://doi.org/10.4064/aa250115-14-7">https://doi.org/10.4064/aa250115-14-7</a>.
  ieee: C. Elsholtz, I. Z. Ruzsa, and L. Wurzinger, “Sumset growth in progression-free
    sets,” <i>Acta Arithmetica</i>, vol. 220. Institute of Mathematics, pp. 289–303,
    2025.
  ista: Elsholtz C, Ruzsa IZ, Wurzinger L. 2025. Sumset growth in progression-free
    sets. Acta Arithmetica. 220, 289–303.
  mla: Elsholtz, Christian, et al. “Sumset Growth in Progression-Free Sets.” <i>Acta
    Arithmetica</i>, vol. 220, Institute of Mathematics, 2025, pp. 289–303, doi:<a
    href="https://doi.org/10.4064/aa250115-14-7">10.4064/aa250115-14-7</a>.
  short: C. Elsholtz, I.Z. Ruzsa, L. Wurzinger, Acta Arithmetica 220 (2025) 289–303.
corr_author: '1'
date_created: 2025-11-04T14:33:16Z
date_published: 2025-09-12T00:00:00Z
date_updated: 2025-12-01T15:18:09Z
day: '12'
department:
- _id: TiBr
doi: 10.4064/aa250115-14-7
external_id:
  isi:
  - '001570716800001'
intvolume: '       220'
isi: 1
language:
- iso: eng
month: '09'
oa_version: None
page: 289-303
publication: Acta Arithmetica
publication_identifier:
  eissn:
  - 1730-6264
  issn:
  - 0065-1036
publication_status: published
publisher: Institute of Mathematics
quality_controlled: '1'
scopus_import: '1'
status: public
title: Sumset growth in progression-free sets
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 220
year: '2025'
...
---
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. <i>The Quarterly Journal
    of Mathematics</i>. 2024;75(4):1243-1254. doi:<a href="https://doi.org/10.1093/qmath/haae044">10.1093/qmath/haae044</a>
  apa: Elsholtz, C., &#38; Wurzinger, L. (2024). Sumsets in the set of squares. <i>The
    Quarterly Journal of Mathematics</i>. Oxford University Press. <a href="https://doi.org/10.1093/qmath/haae044">https://doi.org/10.1093/qmath/haae044</a>
  chicago: Elsholtz, Christian, and Lena Wurzinger. “Sumsets in the Set of Squares.”
    <i>The Quarterly Journal of Mathematics</i>. Oxford University Press, 2024. <a
    href="https://doi.org/10.1093/qmath/haae044">https://doi.org/10.1093/qmath/haae044</a>.
  ieee: C. Elsholtz and L. Wurzinger, “Sumsets in the set of squares,” <i>The Quarterly
    Journal of Mathematics</i>, 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.” <i>The
    Quarterly Journal of Mathematics</i>, vol. 75, no. 4, Oxford University Press,
    2024, pp. 1243–54, doi:<a href="https://doi.org/10.1093/qmath/haae044">10.1093/qmath/haae044</a>.
  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'
...