Sumsets in the Hypercube

TL;DR

This work determines the asymptotic count of sumsets in the Boolean hypercube $\mathbb{F}_2^n$, proving $|\mathcal{S}_n| \sim (2^n-1) 2^{N/2}$ with $N=2^n$, and shows that almost all sumsets contain a complete codimension-1 subspace (i.e., $|\mathcal{S}_n\Delta\mathcal{H}_n| = o(|\mathcal{S}_n|)$). The authors develop structural tools connecting unions of sumsets to independent sets in the hypercube, establish a tight upper bound by concentrating sumsets in $\mathcal{H}_n$, and provide a lower bound via an explicit construction. They also analyze how random subsets of abelian groups decompose into unions of sumsets, proving near-tight bounds on the minimum and typical numbers of required sumsets. The results bridge additive combinatorics with probabilistic methods and have implications for understanding typical structures in finite field models and relevant computational settings.

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 $1$.

Theorems & Definitions (43)

• Theorem 1
• Theorem 2
• Lemma 3
• proof
• Proposition 4
• proof
• Proposition 5
• proof
• Theorem 6
• Theorem 7