@article{20603,
  abstract     = {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.},
  author       = {Elsholtz, Christian and Ruzsa, Imre Z. and Wurzinger, Lena},
  issn         = {1730-6264},
  journal      = {Acta Arithmetica},
  pages        = {289--303},
  publisher    = {Institute of Mathematics},
  title        = {{Sumset growth in progression-free sets}},
  doi          = {10.4064/aa250115-14-7},
  volume       = {220},
  year         = {2025},
}

@article{18930,
  abstract     = {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).},
  author       = {Elsholtz, Christian and Wurzinger, Lena},
  issn         = {1464-3847},
  journal      = {The Quarterly Journal of Mathematics},
  number       = {4},
  pages        = {1243--1254},
  publisher    = {Oxford University Press},
  title        = {{Sumsets in the set of squares}},
  doi          = {10.1093/qmath/haae044},
  volume       = {75},
  year         = {2024},
}