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arXiv · 2403.16589

Sumsets in the Hypercube

Noga Alon·Or Zamir
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Abstract

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 $1$.

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Noga Alon, Or Zamir. 2024-04-16. Sumsets in the Hypercube. https://arxiv.org/abs/2403.16589

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