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Random Cayley sum hypergraphs and k-fold sumsets

  • Jihyo Chae
  • Hyunwoo Lee
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Abstract

We denote by fk(Γ) the largest integer with the property that every subset of a finite abelian group Γ of size at least |Γ| − fk(Γ) is a k-fold sumset. Extending a recent result of Alon and Pham, we prove that fk(Γ) ≤ Õ (n(2k−1)/(4k−3)) holds for all finite abelian groups Γ and integers k ≥ 2, where n = |Γ|. Additionally, we also show that the lower bound fk(Γ) ≥ Ω̃ (n1/k) holds if Γ has no nontrivial element of order dividing k. Our upper bound improves a previous result of Balogh, Liu, and Sharifzadeh, and recovers the bound of Alon and Pham in the case k = 2. The proof relies on a new upper bound for the independence number of random Cayley sum hypergraphs, which may be of independent interest.

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Cite

BibTeXRIS
BibTeX
@misc{chae2026random,
  title = {{Random Cayley sum hypergraphs and $k$-fold sumsets}},
  author = {Jihyo Chae and Hyunwoo Lee},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.08629},
  eprint = {2610.08629},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.08629},
}
RIS
TY  - UNPB
TI  - Random Cayley sum hypergraphs and k-fold sumsets
AU  - Jihyo Chae
AU  - Hyunwoo Lee
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2610.08629
UR  - https://arxiv.org/abs/2610.08629
AB  - We denote by f_k(Γ) the largest integer with the property that every subset of a finite abelian group Γ of size at least |Γ| − f_k(Γ) is a k-fold sumset. Extending a recent result of Alon and Pham, we prove that f_k(Γ) ≤ Õ (n^(2k−1)/(4k−3)) holds for all finite abelian groups Γ and integers k ≥ 2, where n = |Γ|. Additionally, we also show that the lower bound f_k(Γ) ≥ Ω̃ (n^1/k) holds if Γ has no nontrivial element of order dividing k. Our upper bound improves a previous result of Balogh, Liu, and Sharifzadeh, and recovers the bound of Alon and Pham in the case k = 2. The proof relies on a new upper bound for the independence number of random Cayley sum hypergraphs, which may be of independent interest.
ER  -