Sumset growth in progression-free sets Elsholtz, Christian Ruzsa, Imre Z. Wurzinger, Lena ; https://orcid.org/0009-0004-5360-0074 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 |A+B|≥|A|2k−13k−2|B|k3k−2. As 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. Institute of Mathematics 2025 info:eu-repo/semantics/article doc-type:article text http://purl.org/coar/resource_type/c_2df8fbb1 https://research-explorer.ista.ac.at/record/20603 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> eng info:eu-repo/semantics/altIdentifier/doi/10.4064/aa250115-14-7 info:eu-repo/semantics/altIdentifier/issn/0065-1036 info:eu-repo/semantics/altIdentifier/e-issn/1730-6264 info:eu-repo/semantics/altIdentifier/wos/001570716800001 info:eu-repo/semantics/closedAccess