{"publication_identifier":{"issn":["0065-1036"],"eissn":["1730-6264"]},"publication_status":"published","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","volume":220,"intvolume":"       220","publisher":"Institute of Mathematics","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","day":"12","date_created":"2025-11-04T14:33:16Z","type":"journal_article","date_published":"2025-09-12T00:00:00Z","article_type":"original","_id":"20603","article_processing_charge":"No","doi":"10.4064/aa250115-14-7","corr_author":"1","date_updated":"2025-11-04T15:06:03Z","title":"Sumset growth in progression-free sets","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":[{"first_name":"Christian","full_name":"Elsholtz, Christian","last_name":"Elsholtz"},{"first_name":"Imre Z.","full_name":"Ruzsa, Imre Z.","last_name":"Ruzsa"},{"full_name":"Wurzinger, Lena","orcid":"0009-0004-5360-0074","id":"50c57d72-32a8-11ee-aeea-d652094d2ccd","last_name":"Wurzinger","first_name":"Lena"}],"quality_controlled":"1","oa_version":"None","scopus_import":"1","status":"public","language":[{"iso":"eng"}],"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>.","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>","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>.","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>","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.","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."},"OA_type":"closed access","year":"2025","department":[{"_id":"TiBr"}],"month":"09","publication":"Acta Arithmetica","page":"289-303"}