[{"abstract":[{"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.","lang":"eng"}],"author":[{"full_name":"Elsholtz, Christian","first_name":"Christian","last_name":"Elsholtz"},{"first_name":"Imre Z.","last_name":"Ruzsa","full_name":"Ruzsa, Imre Z."},{"full_name":"Wurzinger, Lena","orcid":"0009-0004-5360-0074","first_name":"Lena","last_name":"Wurzinger","id":"50c57d72-32a8-11ee-aeea-d652094d2ccd"}],"page":"289-303","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","year":"2025","isi":1,"publication_identifier":{"eissn":["1730-6264"],"issn":["0065-1036"]},"article_processing_charge":"No","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","_id":"20603","date_published":"2025-09-12T00:00:00Z","quality_controlled":"1","type":"journal_article","oa_version":"None","month":"09","publication_status":"published","doi":"10.4064/aa250115-14-7","date_updated":"2025-12-01T15:18:09Z","title":"Sumset growth in progression-free sets","publication":"Acta Arithmetica","intvolume":"       220","external_id":{"isi":["001570716800001"]},"citation":{"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>.","short":"C. Elsholtz, I.Z. Ruzsa, L. Wurzinger, Acta Arithmetica 220 (2025) 289–303.","ista":"Elsholtz C, Ruzsa IZ, Wurzinger L. 2025. Sumset growth in progression-free sets. Acta Arithmetica. 220, 289–303.","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.","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>","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>.","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>"},"department":[{"_id":"TiBr"}],"corr_author":"1","scopus_import":"1","day":"12","language":[{"iso":"eng"}],"article_type":"original","volume":220,"publisher":"Institute of Mathematics","date_created":"2025-11-04T14:33:16Z","OA_type":"closed access","status":"public"},{"page":"1243-1254","author":[{"last_name":"Elsholtz","first_name":"Christian","full_name":"Elsholtz, Christian"},{"id":"50c57d72-32a8-11ee-aeea-d652094d2ccd","last_name":"Wurzinger","first_name":"Lena","full_name":"Wurzinger, Lena","orcid":"0009-0004-5360-0074"}],"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","short":"CC BY (4.0)"},"abstract":[{"lang":"eng","text":"We study sumsets 𝒜 + ℬ in the set of squares 𝒮 (and, more generally, in the set of kth powers 𝒮k, where k ≥2 is an integer). It is known by a result of Gyarmati that 𝒜 + ℬ ⊂ 𝒮k ∩[1,N] implies that min(|𝒜|,|ℬ|) =Ok(logN). Here, we study how the upper bound on |ℬ| decreases, when the size of |𝒜| increases (or vice versa). In particular, if |𝒜| ≥ 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 |𝒜| ≥ CkloglogN,then |ℬ| = Ok(logN(loglogN)2)."}],"issue":"4","year":"2024","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.","oa":1,"publication_identifier":{"issn":["0033-5606"],"eissn":["1464-3847"]},"isi":1,"article_processing_charge":"Yes (via OA deal)","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_published":"2024-12-01T00:00:00Z","has_accepted_license":"1","_id":"18930","type":"journal_article","quality_controlled":"1","publication_status":"published","month":"12","oa_version":"Published Version","date_updated":"2025-12-04T14:46:28Z","title":"Sumsets in the set of squares","doi":"10.1093/qmath/haae044","external_id":{"isi":["001304396600001"]},"intvolume":"        75","publication":"The Quarterly Journal of Mathematics","file_date_updated":"2025-01-28T07:03:51Z","department":[{"_id":"TiBr"}],"citation":{"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>","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>.","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>","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.","short":"C. Elsholtz, L. Wurzinger, The Quarterly Journal of Mathematics 75 (2024) 1243–1254.","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>."},"corr_author":"1","file":[{"file_id":"18931","file_size":424645,"file_name":"2024_QuarterlyJourMath_Elsholtz.pdf","checksum":"1a06e052761d3f1e873463d6f529dd82","relation":"main_file","date_created":"2025-01-28T07:03:51Z","date_updated":"2025-01-28T07:03:51Z","access_level":"open_access","creator":"dernst","success":1,"content_type":"application/pdf"}],"OA_place":"publisher","scopus_import":"1","article_type":"original","ddc":["510"],"day":"01","language":[{"iso":"eng"}],"publisher":"Oxford University Press","volume":75,"status":"public","OA_type":"hybrid","date_created":"2025-01-28T06:55:31Z","PlanS_conform":"1"}]