Sumsets in the Hypercube
<|reference_start|>Sumsets in the Hypercube: A subset $S$ of the Boolean hypercube $\mathbb{F}_2^n$ is a sumset if $S = A+A = \{a + b \ | \ a, b\in A\}$ for some $A \subseteq \mathbb{F}_2^n$. We prove that the number of sumsets in $\mathbb{F}_2^n$ is asymptotically $(2^n-1)2^{2^{n-1}}$. Furthermore, we show that the family of sumsets in $\mathbb{F}_2^n$ is almost identical to the family of all subsets of $\mathbb{F}_2^n$ that contain a complete linear subspace of co-dimension $1$.<|reference_end|>
@article{alon2024sumsets,
    title={Sumsets in the Hypercube},
    author={Noga Alon, Or Zamir},
    journal={arXiv preprint arXiv:2403.16589},
    year={2024},
    archivePrefix={arXiv},
    eprint={2403.16589},
    primaryClass={math.CO cs.DM}
}