---
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'
das_tickbox: '0'
date_created: 2025-11-04T14:33:16Z
date_published: 2025-09-12T00:00:00Z
date_updated: 2026-07-29T09:51:25Z
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'
researchdata_availability: no
scopus_import: '1'
status: public
supplementarymaterial: no
title: Sumset growth in progression-free sets
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 220
year: '2025'
...