Random Cayley sum hypergraphs and $k$-fold sumsets

We denote by $f_k(\mathrm{\Gamma })$ the largest integer with the property that every subset of a finite abelian group $\mathrm{\Gamma }$ of size at least $\mathrm{\mid }\mathrm{\Gamma }\mathrm{\mid }-f_k(\mathrm{\Gamma })$ is a $k$-fold sumset. Extending a recent result of Alon and Pham, we prove that $$f_k(\mathrm{\Gamma })\le \tilde{O}\left(n^{(2k-1)\mathrm{/}(4k-3)}\right)$$ holds for all finite abelian groups $\mathrm{\Gamma }$ and integers $k\ge 2$, where $n=\mathrm{\mid }\mathrm{\Gamma }\mathrm{\mid }$. Additionally, we also show that the lower bound $f_k(\mathrm{\Gamma })\ge \tilde{\mathrm{\Omega }}\left(n^{1\mathrm{/}k}\right)$ holds if $\mathrm{\Gamma }$ 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.